a little timeline + concepts of things karone is doing after the end of lost galaxy:
seeking atonement→ perhaps the most obvious and predictable one, but i do think it's worth acknowledging that karone would continue to seek out opportunities to make amends for her past as astronema and pay retribution to those she wronged and hurt (if they're still alive😬)
off duty→ we know kendrix takes the quasar saber one last time to resume her role after the end of the season to aid in the lightspeed rescue crossover. which leads me to the next point:
with andros→ at that time of trakeena's revenge arc in lightspeed rescue, karone is spending time reconnecting with her brother therefore unavailable, possibly switching between exploring earth with him and being reintroduced to K0-35 and its culture so she can learn where she came from.
full time ranger→ we also know that by the time of the legendary battle, kendrix is retired and karone is formally the current galaxy pink ranger. this makes sense since kendrix is a stem baddie with a full time job while karone is unemployed lmao
imposter syndrome→ i think once kendrix's retirement is official and the pink galaxy ranger role is solely on karone's hands this is when the imposter syndrome would really kick in. up until this point she was joining the team to help, and she loves being a ranger so she was happy to hold onto the position for as long as it was needed. but now that it's really all on her i think she goes through a similar arc as leo since she didn't pull the quasar saber out of its resting place. she wasn't the chosen one for that role and now she's fully responsible for it, not merely holding the space to help.
rebuild the astro megaship→ since becoming a "full time" ranger i think she would try to team a group of qualified rangers + engineers/mechanics to rebuild the astro megaship. this could be a fun concept for a crossover-ish team to hang out for a while and go on missions together (sourcing materials they need for the ship maybe?)
human normalcy→ she would spend some time exploring different subjects and interests, and may pursue some sort of (or equivalent to) higher education. she doesn't really know what she likes and what's "her thing", and neither do i yet at this point but it's about the journey and self exploration ☝️ this could take place on either mironoi, K0-35 or earth obviously. that would be the easiest for plotting things with other muses.
protecting untouched morphers→ much like she did originally with the pink quasar saber, i could see karone getting involved in efforts to find relics made to access the morphin grid (i.e possible morphers) across galaxies (reference video) to prevent them from falling into the hands of evil. this may include tracking down and fighting mercenaries, collectors, or ex-villains who want those morphers to harvest energy from the morphin grid.
search for ecliptor→ karone knows the chances of finding a way to bring back ecliptor are slim to none, but she still feels like she needs to at least try. she thinks if she's worthy of being given a chance to be good then ecliptor deserves that chance too. in spite of being programmed for evil he loved her and cared for her like a caring parent would and that shows, to her, that he has the capacity to grow beyond his programming. so even if she knows it’s unlikely she dedicates time to tracking down ancient tech, sorcery, or people who might hold the knowledge to restore him somehow. this could lead her to forgotten civilizations, ex-villains willing to help her or even put her at odds with allies who warn her she’s chasing the impossible, or are worried/distrustful of her intentions.