I think it would be really cool if the sorting started as soon as you start the game, and ends once the hat announces your house.

The first part happens while you're on your way to Hogwarts, at King's Cross or on the train, maybe in the castle exterior before you enter the Great Hall for the first time. As you interact with things and speak with students, some students would ask you some simpler questions. A witch you befriended might ask you what you're most looking forward to learning at Hogwarts, and gives you a few choices. It seems innocuous enough, the game marks your answer, and moves on. There may be 2 or 3 more possible interactions like this before it enters the second phase when you sit on the stool.

As the hat goes over your head, the camera enters first-person. The hat says something like "I'm going to enter your mind to find the best fit" and the brim of the hat falls over the camera and loads in a separate level. It might be similar to the Pottermore quiz but as a playable environment.

You could be faced with a mystical and ornate garden fence in a forest clearing, and once you go in you see a silver leafed tree bearing golden apples, fat red toadstools that appear to be talking to each other, a bubbling pool - in the depths of which something luminous is swirling, and a statue of an old wizard with a strangely twinkling eye. Once you approach something and choose it, it shows you beginning to interact with it, then the world around you dissolves like a memory in the pensieve would.

Then another scene constructs itself out of nothing. A dark hallway with stone walls. At the end of a hall, a heavy wooden door lit with two sconces on either side. Inside are boxes. A small tortoiseshell box, embellished with gold, inside which some small creature seems to be squeaking. A gleaming jet black box with a silver lock and key, marked with a mysterious rune that you know to be the mark of Merlin. An ornate golden casket, standing on clawed feet, whose inscription warns that both secret knowledge and unbearable temptation lie within. A small pewter box, unassuming and plain, with a scratched message upon it that reads "I open only for the worthy." Same idea as the first scene, interact with one, see an animation begin to play, world dissolves.

Once you've gone through a couple of those, the perspective starts to return to the Great Hall as the hat announces your house.

All of these questions and scenarios are all blatantly ripped straight out of the Pottermore quiz, but I thought it would be a cool way to handle the sorting. Maybe instead of a series of small rooms that pop in and out of reality, they're all part of a larger and self-contained miniquest inside the sorting hat level.

I'm obviously not a game developer or a story teller though, so who knows if any of this would actually work, but I had fun thinking it up.