These English glosses make it unclear what you're driving at, since 'back' and 'again' are polysemous and overlap.
At any rate, here's my full analysis of πάλιν in the New Testament, in which the πάλιν of Gal 2:1 marks the repetition of Paul's action of going to Jerusualem (§ 2.1 below):
πάλιν back, again
For marking discrete repetition of ① a result state, ② an action, ③ a full event, ④ a location, ⑤ a setting or participant, or ⑥ a discourse unit.
[frozen acc. of zero-grade *πάλις “turning, turn” (cf. o-grade πόλος, “axis, pole”) > ⓪ counter-directional path adverb: πάλιν MOTION-VERB, for a path in the other direction, back(wards). This sense is obsolete in the NT and renewed by such as locutions as εἰς τὰ ὀπίσω (John 6:66 18:6 ἀπῆλθον, John 20:14 ἐστράφη), possibly also with εἰς τὸ πάλιν 2 Cor 13:2 ὅτι ἐὰν ἔλθω εἰς τὸ πάλιν οὐ φείσομαι if I turn around and come back, I will not spare.]
① Restitutive adverb: πάλιν REVERSE-CHANGE, for a change back to a prior ¹location, ²possessor, or ³state.
[< Same entailments as ⓪, as] in travel itineraries: 2 Cor 1:16 (καὶ δι’ ὑμῶν διελθεῖν εἰς Μακεδονίαν) καὶ πάλιν ἀπο Μακεδονίας ἐλθεῖν πρὸς ὑμᾶς and to come back from Macedonia to you. Gal 1:17 (ἀπῆλθον εἰς Ἀραβίαν) καὶ πάλιν ὑπέστρεψα εἰς Δαμασκόν.
¹ πάλιν ἐλθεῖν: John 14:3 (καὶ ἐὰν πορευθῶ ...,) πάλιν ἔρχομαι I am coming back. & other verbs of motion: Mark 7:31 Καὶ πάλιν ἐξελθὼν ἐκ τὸν ὀρίων Τύρου (cf. v.24 ἀπῆλθεν εἰς τὰ ὅρια Τύρου) And coming back from the regions of Tyre. • In hyperbaton: 2 Cor 2:1 τὸ μὴ πάλιν ἐν λύπῃ πρὸς ὑμᾶς ἐλθεῖν to not come back to you in pain. • With clarifying GOAL: John 11:8 καὶ πάλιν ὑπάγεις ἐκεῖ; And you’re going back, there? John 18:38 Καὶ τοῦτο εἰπὼν, πάλιν ἐξῆλθεν πρὸς τοὺς Ἰουδαίους.
² πάλιν λαβεῖν: John 10:17 (ἐγὼ τίθημι τὴν ψυχήν μου) ἵνα πάλιν λάβω αὐτήν to take it back. John 10:18 (ἐξουσίαν ἔχω θεῖναι αὐτήν) καὶ ἐξουσίαν ἔχω πάλιν λαβεῖν αὐτήν.
³ With other change-of-state verbs (in context with a salient opposite change): Gal 2:18 (ἃ κατέλυσα) ταῦτα πάλιν οἰκοδομῶ I build them again. Rom 11:23 δυνατὸς γάρ ἐστιν ὁ θεός πάλιν ἐγκεντίσαι αὐτούς for God can graft them back in (cf. v.20 τῇ ἀπιστίᾳ ἐξεκλάσθησαν). 2 Pet 2:20 (ἀποφυγόντες τὸ μιάσματα τοῦ κόσμου) τούτοις δὲ πάλιν ἐμπλακέντες getting tangled up again in them. • Emphatic with ἀνα-prefixed verbs: Acts 18:21 Πάλιν ἀνακάμψω πρὸς ὑμᾶς I will return back to you. Heb 6:6 (καὶ παραπεσόντας,) πάλιν ἀνακαινίζειν εἰς μετάνοιαν to renew them back into repentance.
② Repetitive adverb: πάλιν VERB, for repetition of an action that occurred before, with ¹verbs of change, ²verbs of saying and speaking, and ³other activity verbs.
¹ πάλιν CHANGE-VERB [< ①] with contextually salient same change: Mark 8:13 πάλιν ἐμβάς got back in the boat > got in the boat again (cf. v.10 ἐμβᾶς εἰς τὸ πλοῖον). Gal 2:1 Ἔπειτα διὰ ιδ΄ ἐτῶν, πάλιν ἀνέβην εἰς Ἱεροσόλυμα I went back/again to Jerusalem (cf. 1:18 Ἔπειτα μετὰ ἔτη τρίας ἀνῆλθον εἰς Ἱεροσόλυμα). Heb 6:1 (ἀφέντες τὸν τῆς άρχῆς τοῦ Ἰησοῦ λόγον ...) μὴ πάλιν θεμέλιον καταβαλλόμενοι. • Without a contextually salient opposite change: John 8:2 πάλιν παρεγένετο εἰς τὸ ἱερόν he appeared again in the temple > he reappeared in the temple. Heb 1:6 ὅταν δὲ πάλιν εἰσαγάγῃ τὸν πρωτότοκον εἰς τὴν οἰκουμένην But when he brings the firstborn into the world again.
² πάλιν εἰπεῖν, to speak again, to repeat: Mark 14:69 ἤρξατο πάλιν λέγειν τοῖς παρεστῶτος she began to repeat to the bystanders. John 18:40 έκραύγασαν οὖν πάλιν λέγοντες so they cried out while repeating. Gal 1:9 ὡς προειρήκαμεν καὶ ἄρτι πάλιν λέγω as I have declared and now I repeat. Phil 4:4 Πάλιν ἐρῶ «». & other verbs of speaking: John 8:12 Πάλιν οὖν αὐτοῖς ἐλάλησεν ὁ Ἰησοῦς. Rev 10:8 πάλιν λαλοῦσας μετ’ ἐμοῦ. Mark 10:1b καὶ ὡς εἰώθει πάλιν ἐδίδασκεν αὐτούς. John 12:28 καὶ πάλιν δοξάσω. Matt 26:72 καὶ πάλιν ἠρνήσατο μετὰ ὄρκου ὅτι «» (oath not in the scope of the repetition). John 18:27 πάλιν οὖν ἡρνήσατο Πέτρος. Jas 5:18 καὶ πάλιν προσηύξατο. Rev 10:11 Δεῖ σε πάλιν προφητεῦσαι. 2 Cor 3:1 Ἀρχόμεθα πάλιν ἑαυτοὺς συνιστάνειν; 2 Cor 5:12 οὐ πάλιν ἑαυτοὺς συνιστάνομεν ὑμῖν. Heb 4:7 πάλιν τινὰ ὁρίζει ἡμέραν. Matt 27:50 ὁ δὲ Ἰησοῦς πάλιν κράξας φωνῇ μεγάλῃ (cf. v.46). Gal 4:19 πάλιν ὠδίνω.
³ With other verbs: John 9:27 τί πάλιν θέλετε ἀκούειν; Why do you want to hear it again? John 11:38 Ιησους οὖν πάλιν ἐμβριμώμενος ἐν ἑαυτῷ. John 16:22 πάλιν δὲ ὄψομαι ὑμᾶς. Phil 2:28 ἵνα ἰδόντες αὐτὸν, πάλιν χαρῆτε, κἀγὼ ἀλυπότερος ὦ so when you see him, you will be happy again and I less anxious. • With effected group: Mark 8:1 πάλιν πολλοῦ ὄχλου ὄντος καὶ μὴ ἐχόντων τί φάγωσιν.
③ Clause-final iterative marker: SITUATION πάλιν, once again, once more, for repetition of the whole situation, especially for multiple times.
¹ With verbs of movement: John 11:7 Ἄγωμεν εἰς τὴν Ἰουσαίαν πάλιν Let us go to Judea once more. John 13:12 καὶ ἀνέπεσεν πάλιν and he reclined (to dine) once more. John 19:9 καὶ εἰσῆλθεν εἰς τὸ πραιτώριον πάλιν and he entered the pretorium once again (cf. 18:33 Εἰσῆλθεν οὖν πάλιν εἰς τὸ πραιτώριον). & other change of states: John 4:13 (Πᾶς ὁ πίνων ἐκ τοῦ ὕδατος τούτου) διψήσει πάλιν will thirst yet again.
² With verbs of saying, esp. with repeated content: Matt 26:44b τὸν αὐτὸν λόγον εἰπὼν πάλιν saying the same thing once again. John 9:17 λέγουσιν οὖν τῷ τυφλῷ πάλιν «» so they said to the blind man once again. John 20:21 εἶπεν οὖν αὐτοῖς [ὁ Ἰησοῦς] πάλιν «Εἰρήνη ὑμῖν» so Jesus told them once more “Peace to you.”
³ Pleonastic for emphasis: Acts 17:32 Ἀκουσόμεθά σου περὶ τούτου, καὶ πάλιν We will hear about this from you one more time. John 21:16 λέγει αὐτῷ, πάλιν δεύτερον He told him again a second time.
④ Repetitive focus particle: πάλιν XP, for presupposing a prior occurrence of a location or state, in various prosodic contexts: ¹post-nuclear elaboration, ²adnominal, ³preverbal locative.
¹ …, πάλιν XP: Emphatic in the right periphery for a post-nuclear elaboration: Rom 8:15 οὐ γὰρ ἐλάβετε πνεῦμα δουλείας, πάλιν εἰς φόβον. Gal 4:9a πῶς ἐπιστρέφετε, πάλιν ἐπὶ τὰ ἀσθενῇ καὶ πτωχὰ στοιχεῖα. Mark 11:3 καὶ εὐθὺς αὐτὸν ἀποστέλλει, πάλιν ὧδε. • With dative: 2 Cor 10:7 (εἴ τις πέποιθεν ἑαυτῷ ...) τοῦτο λογιζέσθω, πάλιν ἐφ’ ἑαυτῷ ὅτι «».
² πάλιν LOC: Adnominal in Phil 1:26 διὰ τῆς ἐμῆς παρουσίας πάλιν πρὸς ὑμᾶς through my presence again with you. John 16:16, 17, 19 καὶ πάλιν μικρὸν (καὶ ὄψεσθέ με) and after a little while more (temporal accusative) • With second-position copula (cf. ②): John 20:26 πάλιν ἦσαν ἔσω οἱ μαθηταὶ αὐτοῦ his disciples were (outside again >) back outside.
³ πάλιν LOC VERB: Fronted in 1 Cor 7:5 καὶ πάλιν ἐπὶ τὸ αὐτὸ ἦτε and be in the same place again. Gal 4:9b οἷς πάλιν ἄνωθεν δουλεύειν θέλετε; whom you want to serve all over again. Gal 5:1 καὶ μὴ πάλιν ζυγῷ δουλείας ἐνέχεσθε and not be subject again to a yoke of slavery.
⑤ Reframing topical particle: POINT-OF-DEPARTURE πάλιν ..., in second-position for re-establishing ¹a prior setting, ²another line of discourse, ³another time interval, or ⁴another participant’s turn at speaking.
¹ VERB πάλιν LOC < ④ in frame-setting expressions: Post-verbal focus on locations for narrative scene-shifting: Mark 2:1 Καὶ εἰσελθὼν πάλιν εἰς Καφαρναούμ. Mark 2:13 Καὶ ἐξῆλθεν πάλιν παρὰ τὴν θάλασσαν. Mark 3:1 Καὶ είσῆλθεν πάλιν εἰς τὴν συναγωγήν. Mark 11:27 Καὶ ἔρχονται πάλιν εἰς Ἰεροσόλυμα. John 4:3 καὶ ἀπῆλθεν πάλιν εἰς τὴν Γαλιλαίαν. John 4:46 Ἦλθεν οὖν πάλιν εἰς τὴν Κανὰ τῆς Γαλιλαίας. John 6:15 άνεχώρησεν πάλιν εἰς τὸ ὄρος, αὐτὸς μόνος. John 10:40 Καὶ ἀπῆλθεν πάλιν πέραν τοῦ Ἰορδάνου. John 18:33 Εἰσῆλθεν οὖν πάλιν εἰς τὸ πραιτώριον ὁ Πιλᾶτος. John 19:4 καὶ ἐξῆλθεν πάλιν ἔξω ὁ Πιλᾶτος. John 20:10 ἀπῆλθον οὖν πάλιν πρὸς αὐτοὺς οἱ μαθηταί. • In a frame-setting participial: Mark 5:21 Καὶ διαπεράσαντος τοῦ Ἰησοῦς πάλιν εἰς τὸ πέραν (cf. ④). • With re-formed groups and effected objects: Mark 3:20 καὶ συνέρχεται πάλιν ὄχλος. Mark 7:14 Καὶ προσκαλεσάμενος πάλιν τὸν ὄχλον. Mark 10:1a καὶ συμπορεύονται πάλιν ὄχλοι πρὸς αὐτὸν. Mark 10:32 καὶ παραλαβὼν πάλιν τοὺς δώδεκα. Acts 10:15 καὶ φωνὴ πάλιν ἐκ δευτέρου πρὸς αὐτόν. • With addressee in argumentative meta-comment: Gal 5:3 μαρτύρομαι δὲ πάλιν παντὶ ἀνθρώπῳ περιτεμνομένῳ «» (cf. v.2 λέγω ὑμῖν ὅτι ἐὰν περιτέμνησθε «».). • With bare locative: Mark 10:10 Καὶ εἰς τὴν οἰκίαν πάλιν, οἱ μαθηταὶ περὶ τούτου ἐπηρώτων αὐτόν. And back home, ….
² With predicate themes to resume mainline after offline material: John 8:21 Εἶπον οὖν πάλιν αὐτοῖς «». John 10:7 Εἶπεν οὖν πάλιν ὁ Ἰησοῦς «». John 21:1 ἐφανέρωσεν ἑαυτὸν πάλιν ὁ Ἰησοῦς τοῖς μαθηταῖς Jesus showed himself again to the disciples. Acts 11:10 καὶ ἀνεσπάσθη πάλιν ἅπαντα εἰς τὸν οὐρανόν (cf. v.5 καὶ καταβαῖνον σκεῦός τι). 2 Cor 12:21 (v.20 φόβομαι ...) μὴ πάλιν, ἐλθόντος μου ταπεινώσῃ ὁ θεός μου πρὸς ὑμᾶς. • For resuming offline material: John 4:54 Τοῦτο [δὲ] πάλιν, δεύτερον σημεῖον ἐποίησεν ὁ Ἰησοῦς. Relating to Jesus’s antagonists: John 10:19 Σχίσμα πάλιν ἐγένετο ἐν τοῖς Ἰουδαίοις. John 10:31 Ἐβάστασαν πάλιν λίθους οἱ Ἰουδαῖοι. John 10:39 Ἐζήτουν [οὖν] αὐτὸν πάλιν πιάσαι So they were trying again to arrest him.
³ TIME πάλιν, …: Repeated time interval, after that: Mark 14:70b καὶ μετὰ μικρὸν πάλιν, οἱ πάρεστῶτες ἔλεγον τῷ Πέτρῳ And a little while after that. John 1:35 Τῇ ἐπαύρον πάλιν εἰστηκει ὁ Ἰωάννης. On the next day after that.
⁴ ὁ δὲ (PARTICIPANT) πάλιν, …: Returning to a participant for another turn (cf. ⑥²), in turn: Mark 10:24 ὁ δὲ Ἰησοῦς πάλιν ἀποκριθεὶς λέγει «» (repeated content). Mark 14:70a ὁ δὲ πάλιν ἠρνεῖτο (repeated denial). Mark 15:4 ὁ δὲ Πιλᾶτος πάλιν, ἐπηρώτα αὐτὸν λέγων «» (cf. v2, different question). Mark 15:12 ὁ δὲ Πιλᾶτος πάλιν ἀποκριθεὶς, ἔλεγεν αὐτοῖς «». Mark 15:13 οἱ δὲ πάλιν ἔκραξαν «».
⑥ Parallel discourse marker: (καὶ) πάλιν, TEXT: for ¹a logically parallel proposition, ²another citation, or ³a narratively parallel set of actions.
¹ For parallel propositions, often with inversion: Luke 6:43 οὐδὲ πάλιν, δένδρον σαπρὸν ποιοῦν καρπὸν καλόν πάλιν. 1 John 2:8 πάλιν, ἐντολὴ καινὴν γράφω ὑμῖν Then again, it is a new commandment that I’m writing to you. John 16:28 (ἐξῆλθον παρὰ τοῦ πατρὸς καὶ ἐλήλυθα εἰς τὸν κόσμον·) πάλιν, ἀφίημι τὸν κόσμον καὶ πορεύομαι πρὸς τὸν πατέρα. Heb 5:12 (καὶ γὰρ ὀφείλοντος εἶναι διδάσκαλοι διὰ τὸν χρόνον) πάλιν, χρείαν ἔχετε τοῦ διδάσκειν ὑμᾶς τινά but then again you need someone to teach you. × Parallel introduction: 1 Cor 12:21 (οὐ δύναται δὲ ὁ ὀφθαλμὸς εἰπεῖν τῇ χειρί «») ἢ πάλιν, ἡ κεφαλὴ τοῖς ποσίν «».
² For further citations: Rom 15:11, 1 Cor 3:20, Heb 1:5 2:13a 13b 10:30 καὶ πάλιν, «» (with omission of the verb). Heb 4:5 καὶ ἐν τούτῳ πάλιν, «» (cf. ⑤). Rom 15:10 καὶ πάλιν, λέγει «». Rom 15:12 καὶ πάλιν, Ήσαΐας λέγει «». John 12:39 ὅτι πάλιν, εἶπεν Ἠσαΐας «». John 19:37 καὶ πάλιν, ἑτερα γραφὴ λέγει «». Matt 4:7 Πάλιν, γέγραπται «» Again, it is written. Matt 5:33 Πάλιν, ἠκούσατε ὅτι ἐρρέθη τοῖς άρχαίος, «» • To introduce another parable: Matt 13:45, 47 Πάλιν, ὁμοία ἐστὶν ἡ βασιλεία τῶν οὐρανῶν Again, the kingdom of heaven is like. For meta-comments: Matt 18:19 Πάλιν, λέγω ὑμῖν «». 2 Cor 11:16 Πάλιν, λέγω «». • More generally, for another turn at speaking (with different content): Matt 19:24 πάλιν δὲ, λέγω ὑμῖν «». Mark 14:61 πάλιν ὁ ἀρχιερεὺς, ἐπηρώτα αὐτόν «». Luke 13:20 Καὶ πάλιν εἶπεν «». Luke 23:20 πάλιν δὲ ὁ Πιλᾶτος, προσεφώνησεν αύτούς «». John 18:7 πάλιν οὖν, ἐπηρώτησεν αὐτούς.
³ For repetitive sub-scenes and other complex actions: Matt 4:8 Πάλιν, παραλαμβάνει αὐτὸν ὁ διάβολος ... Once again, the devil takes him (cf. vv.3-4, τότε vv.5-7). Matt 20:5 πάλιν δὲ, ἐξελθὼν περὶ ἕκτην καὶ ἐνάντην ὥραν (cf. different hour in v.3). Matt 21:36, 22:4 πάλιν, ἀπέστειλεν ἄλλους δούλους ... (cf. 21:34, 22:3). Matt 26:42 πάλιν, ἐκ δευτέρου ἀπελθών προσηύξατο (cf. v.39). Matt 26:43 καὶ ἐλθὼν, πάλιν, εὗρεν αὐτοὺς καθεύδοντας (cf. v.40a). Matt 26:44a καὶ ἀφεὶς αὐτοὺς, πάλιν, ἀπελθὼν προσηύξατο ἐκ τρίτου (cf. vv. 39, 42). Mark 4:1 Καὶ πάλιν, ἤρξατο διδάσκειν παρὰ τὴν θάλασσαν καὶ συνάγεται πρὸς αὐτὸν ὄχλος πλεῖστος (cf. 2:13). Mark 8:25 εἶτα πάλιν, ἐπέθηκεν τὰς χεῖρας ἐπὶ τοὺς ὀφαλμοὺς αὐτοῦ (second healing attempt, cf. v.23). Mark 12:4 καὶ πάλιν, ἀπέστειλεν πρὸς αὐτοὺς ἄλλον δοῦλον ... (cf. v.2). Mark 14:39 καὶ πάλιν, ἀπελθὼν προσηύξατο τὸν αὐτὸν λόγον (cf. v.35). Mark 14:40 καὶ πάλιν, ἐλθῶν εὗρεν αὐτοὺς καθεύδοντας (cf. v.37). John 8:8 καὶ πάλιν, κατακύψας ἔγραφεν εἰς τὴν γῆν (cf. v.6). John 9:15 πάλιν οὖν, ἠρώτων αὐτὸν καὶ οἱ Φαρισαῖοι (cf. different subject in v.8 ὁ γείτονες v.10 ἔλεγον οὖν αὐτῷ). Acts 27:28 (καὶ βολίσαντες εὗρον ὀργυιὰς εἴκοσι) καὶ πάλιν βολίσαντες εὗρον ὀργυιὰς δεκαπέντε (cf. ②).