Though, I guess, since whatshisname built it himself, maybe he just modified it so that it's able to hover?

Hank McCoy/Beast built it in the film, and yeah he could've modified it. Given how long it's been since I read a book, I don't remember how it was built in the comics, but I think it included some alien technology there and could do a lot of stuff an ordinary terrestrial Blackbird could never do.