When Kyon, Mikuru, and Koizumi went to a café after saving Nagato, Mikuru seemed rather distracted and then suggested going to a different café than normal in case Haruhi wanted to contact them, as it would be better if they were closer. Kyon and Adult!Mikuru were going to the normal café at that point, which is why the order was sent.
Why Yuki would only acknowledge Kyon's decision of save her life while she thanked him when he lowered her data manipulation load? Because by that time Yuki didn't give any value to her own life.
In Chapter 47, Nonoko, who was down with fever several minutes ago, destroys hordes of Combine hunters and striders like nothing, using Kyon's tools to shoot Radiant Busters without any care for power usage. Why? Altair and Vega run on user's body heat, so her fever provides then with abundant, unlimited power!
The Augmented Reality-based tactical display in Chapter 61 effectively granted Kyon with X-Ray Vision and Night-Vision Goggles. How come he missed Miyuki hiding within hearing distance? He didn't. Kyon just mistook the "vaguely female shape hiding outside of the cloud" for Kyouko, who he though accessed Kuyou's Shadow Song together with Koizumi.
In Higurashi, every new arc takes place in a new alternate reality; in most of those worlds, the protagonists were unable to stop the Great Hinamizawa Disaster that wiped out the village, including Kyon's mother note for the maths to have checked out, she would have had to be alive at that point - and since Higurashi never really shows characters if they're not plot relevant..., meaning Kyon was never born. In those worlds, Haruhi probably either unhappily accepted reality and lost her powers, never had them in the first place, or wiped out the world in angry boredom. And, in the Chapter 3, Kanae mentioned briefly that the worst worlds she'd been to were the ones where Kyon "wasn't anywhere in the world she could find."
It took this troper reading that twice to get the full force of it. In almost all of the "worst world" universes, Kyon's mother died. In almost all of the universes still around, Kyon is alive.
Not necessarily. First, the "worst worlds" might just be from her perspective. If he's the one thing that stays the same most of the time, not having him around will be disconcerting. Second, just because she can't find him doesn't mean he's not there; he could just be somewhere she doesn't know to look. Remember, this was before she learned to teleport to a specific person within the same world, so just not living in Nishinomiya might be enough. Still not an unlikely interpretation though.
Probably disproved as of Chapter 45. From the fact Kanae can teleport to a known person's location, it was suggested that Kanae unconsciously tracked down Kyon's Alternate Selves whenever she slided into a new world.
Kyon's aunt Rena reacts to the news of Kyon being engaged to Tsuruya by the will of his mother by reflexively closing her right hand. She has to consciously restrain herself with her other hand.
Have you ever wondered why a human like Wataru enjoys of so much freedom of action and authority to call the shots of the Combine's resources? Simple enough, he doesn't. The Mind-Control Device implanted in his brain makes sure he can't work against the Combine, which makes him an potentially unwilling but reliable ally to carry out their objectives in human-populated worlds outside of their direct influence.
Thanks to their multiple visits to Hinamizawa, it's possible that one or more members of the SOS Brigade have Hinamizawa Syndrome.
The cover-up of the All Part of the Show bit was entirely unnecessary. The yakuza? Even Evil Has Standards. The Bavarian Fire Drill? Bystanders would get in the way, so quick thinking to disperse them. The telekinesis, tactical armor, and misc. inexplicables? "I'm as clueless as you are." The motive for the hit? Kyon's Big Damn Hero work a few chapters ago, which was public knowledge. Remember, kids: the best lie looks like the truth.