The difference between young Hibari and adult Hibari is an interesting one, to me. They’re both sublimely confident of winning, both prickly and short-tempered around most people, but adult Hibari seems to be somewhat calmer and more balanced.
Now it’s not difficult to get more balanced than young Hibari, because he strikes me very much as the shounen trope of the crazy-strong character. They’re all over the place. They’re the one who’s never been beaten, who never holds back, who loves to go all out with no holds barred and is a little insane with it. The classic method of dealing with this character is to have the hero defeat them, after which they get a lot saner (PoT’s Akutsu, RK’s Soujirou), or, sometimes, to just leave them as the crazy, ambiguous, dangerous one who’s never been beaten (HxH’s Hisoka).
My own framing of that particular dynamic is that the crazy-strong characters are as if running, down a hill. They’re full tilt and flat out, and they’re a breath away from falling, and nothing can stop it–and they know it. That’s what makes them crazy, that edge of almost-disaster, that lack of balance. When they find an equal opponent, then they have someone to lean on, someone who can push back; that way they can still go all out and not be in danger of falling over from lack of resistance. Having someone capable of standing against them balances them.
And I suspect this is, indeed, what happens to Hibari. The part that interests me is that he doesn’t quite follow the classic curve. We never see him being defeated by anyone except Mukuro, and that does not, of course, do anything to make him less volatile. It does, however, seem to focus him, witness the way Reborn uses the possibility of a rematch to lure Hibari into the family properly.
Immediately after that is the event that I think points toward another part of this equation. Yamamoto tries to talk Hibari down, as he is wont to do, and, when that fails, actually stops him. On the one hand is the physical move, which impresses pretty much everyone, and on the other is the psychological move; Yamamoto claims Squalo as his opponent, owning to an actual will to fight and win (to carnivore-hood, as it were). That, I suspect, fixes Hibari’s attention just as much as the way Yamamoto caught his strike.
This, incidentally, is one reason I think that Hibari’s character really changes between the first story and the second. There is a rather similar moment during the scrap for the sakura-viewing spot, in which Yamamoto interferes and catches Hibari’s strike. At that time, Hibari does not focus on him; indeed, he seems put out, perhaps even petulant. The difference in response on those two occasions is notable. Similarly, in the Mukuro arc, when Gokudera busts Hibari out, Hibari (of course) says he could have done it himself, but he then asks if he can/should get rid of Mukuro’s two henchmen. It’s remarkably polite, in comparison to Hibari pre-60, and indicates at least some awareness of obligation. Arrogantly, to be sure. He even hauls Gokudera up the stairs with him, something that the young Hibari does again when he comes forward in time, once again to discharge the obligation of having been helped out. This suggests that he does acknowledge having been helped, which is very much not the same character we are shown in pre-60.
At any rate, I suspect it’s these two things together, falling invisibly in the ten year gap, that calm Hibari: being able to fight again with Mukuro, apparently his eternal opponent given what he says in the future about honing himself against illusionists, and having Yamamoto around, as irregularly intense as he is, to fight with as a friend and probably to puzzle over as a hobby given how often Yamamoto acts in ways Hibari classifies as herbivorous.
Of course, there’s a third element in here: Dino. I also find it very interesting that we barely see the fights/training between Dino and Hibari at all, even though Dino apparently persuades Hibari to not only keep training but to do it away from Hibari’s precious school. However Dino does it, and we don’t get to find that out, this may form the necessary bridge between the loss to Mukuro, which is both tempering and inflaming, and the theoretical sparring with Yamamoto, who I speculate both interests and frustrates Hibari because Yamamoto exists outside the rules of Hibari’s worldview (more on that later). Between these three points, and taking into account my theory that Hibari transfers his love for the institution of the school to the institution of the family, we have a possible explanation for future Hibari, who is calmer and capable of cooperation–possibly even with Mukuro, witness Mukuro’s routing of information to Hibari.
Alternatively, of course, we could put it all down to Tsuna’s powers of redemption as the pure-hearted hero, but those haven’t been leaned on hard enough for me to call that the only reason.