Here's something a bit more involved than other suggestions: I'd like to see Kirk evolve more into the Kirk we're familiar with, which means the character needs to lose the sarcastic smartass persona and start to embrace that oh-so-Kirkian dead seriousness, verging on the corny.
This is easy enough to explain away in the current timeline if the smartass persona was always a defense mechanism that you'd expect in a young and callow person. Once he starts to grow up, he'll develop the self-confidence to let the defenses fall away. There was always a weirdly vulnerable aspect to Shatner's Kirk - he just put himself way out there, without worrying about consequences - it gave him the aura of a truly courageous person, in a way you rarely see in either fiction or real life. He's certainly physically courageous, but to add emotional courage to the mix elevates him to another level.
People usually go through life swaddled in personal defenses, but not our Jim Kirk. That high-wire act of his was always a big part of the character's appeal (and helped set him off from Spock, who of course is Mr. Personal Defenses.)
Just to clarify, I don't mean that Kirk should lose his sense of humor. Smartass-ness is not the same thing. I want him to lose the type of cowardice that stems from immaturity.