The downside about the AT is that overall it's not very scenic; it offers a lot, but IMO doesn't hold up well to all sorts of other types of trails in the U.S.
Early May certainly does limit you, however (including on some parts of the AT). The suggestion of New Mexico is a good one, and particularly the Gila River area --- fun stuff. You might look around Utah as well. Early May is a standard time for thru-hikers to be in Southern California on the PCT, FWIW. I liked the Gila River better, but the PCT offers a lot of infrastructure to go whatever amount of time and per-day mileage you want.
Where I live (Washington State), if we want to hike in early May we either think of going well east of the Cascade Mountains, or hiking the coast, typically the Olympic Coast. Coast hiking is somewhat unique, offering unique challenges, but is pretty darned cool too. My wife and I did a 5-day hiking trip from the Hoh River to Cape Alava on the Olympic Coast a few years back, in April as I recall, and it was an absolute hoot. One can also hike along the coast in Oregon; I've done a few days of that as well, but logistics might be more challenging (?). There's also the West Coast trail in the Vancouver B.C. area,
https://en.wikipedia.org/wiki/West_Coast_Trail Again in my neck of the woods, more or less, some like to do a PCT loop starting and ending at the lowest point on the PCT --- Cascade Locks, at the Oregon/Washington border. The loop follows the PCT and another (more fun) trail called Eagle Creek, and all of this should be fine in early May, perhaps depending on the particular year (?). It really depends on what kind of hiking experience you're looking for.
Bottom line is that in early May I think you go for low elevation (including coastal hiking), or relatively southern states. Ironically, I think you're too LATE in the year for the Florida Trail, but there are a whole lot of somewhat early season options I think, on a state-by-state basis.