From part of that same Alice fic I was working on but put to the side to work on LT3 things.
Alice is not as easy to find as Hatter thinks. He rather expected he'd waltz through the Glass, take two steps and find himself at Alice's door, easy as anything. He steps out instead into an alley and that's where he finds her lying. He carries her out and is stunning by the differences between her world and his. He manages to get help, get her to a hospital, but before he can wait for her to wake, he's pushed away, family only they say, and he can't get in to see her.
They won't tell him where she lives.
There are a lot of people named Alice in her world, and none of the ones he finds the day after she gets released are the right one.
Hatter learns that surnames are important in this world; they distinguish people with the same given name. He realizes he doesn't know hers.
Much as he hates doing it, he goes back to Wonderland, visits Jack. Jack's King now with his hands so very busy, full of his kingdom and of the duchess. Hatter's only too eager to leave them both to their devices after he learns he has to look for Alice Hamilton. He begrudgingly accepts advice from Jack and ventures back to Alice's world.
He hopes this is the last time he has to go back to Wonderland, unless he's going with Alice.