Post-game being after the whole Zero-Lab scene and everything.
The only thing I could find is battling the gym leaders again, which seems like a waste of time.
After beating the gyms you do a tournament where you get a champion rematch, along with winning letting you go to Penny, Arven, and Nemona's dorms. Then you can do the tournament as many times as you want, and winning gives valuable reward like bottle caps and unobtainable nature mints. You also unlock higher star raid battles, along with sidequests for black crystal six star raids.
Source: Beating the game twice.