The limiting factor going south to north is that you have to climb Katahdin in Baxter State Park at the northern end. And you probably want to do this by about October 15 (and will run into park restrictions shortly after that). (There is a lot of steep rocky terrain through northern NH and areas of Maine as well).
So you have to work back from that to when you start from Springer Mountain in GA which can be as early as about March 15 I gather although you're still sort of semi-winter conditions. Some people recommend starting more like mid-April for better weather and fewer through-hiker crrowdsbut you're obviously giving up 30 days or so of margin.