If Gamefreak want to know what to put in their postgame they should look even as far back as Pokemon Platinum. THAT was a brilliant post-game filled with tones of content, and then they managed to overshadow that with HGSS and BW2 too!!
Obviously I wanna see the Battle Frontier or PWT, if not, some other kind of surprise Battle Facility filled with tones of new options and possibilities to explore, including a rental battle mode at least.
If no PWT, I definitely want them to expand the Battle Tree. Give us more characters, even from Sun and Moon, throw Gladion and Faba in there, bring back more old favourites like Silver, Steven, Cheren, hell even villianous bosses like Maxie and Archie (I don't care if it doesn't make sense), let us battle Blue in the single battle mode too, and most of all no dang version exclusive trainers. Throw in new modes too, like PWT had, and give us a mode to just battle famous trainers.
I'd love to be able to fight all the Trial captains post game, return to Lana/Kiawe/Mallow's houses and challenge them to a fight (or rematch). Give us more trainers who challenge you in the Pokemon League (oh, and give them 6 Pokemon), give some of them Megas as well and that applies to the E4 too.
I like the idea some people have where after the post game, Gyms start to crop up, I would love to be able to battle 8 Gym Leaders in the post game, with a special reward if you beat them all (maybe this would unlock the Battle Frontier). If we do get Gym Leaders it would be neat if some of them happen to be Gym Leaders from other regions who moved.
All in all, while I don't expect most of this stuff to happen, Game Freak have never seemed to dissapoint when it comes to third versions, since they don't have to start everything from scratch again they have more time to give to vets and addicts like us who appreciate the goldmines of content many Pokemon games can give, this is why I'm really looking forward to USUM, I know I'm gonna have a blast going through an expanded and updated Alola region, especially since I did the first time around.