The only pinch is that the Big K isn’t open in those months. Usually opens late May after the thaw and Baxter Park camping closes October 15th. It would be a brutal fifteen from outside the park in harsh weather, not to mention the five back down the mountain. I think the rest of the trail is open most of the year, though parts (like the Whites) are still quite intimidating.
You would also have to consider that snow hiking is a lot harder and thus, slower, than fair weather hiking, so you might want to allow yourself more time (seven months? eight?) And there are fewer hikers out. It would be a whole different trail. (Which could be good, depending on what you are going for…)
I met a guy, Gizmo, southbound as I was starting out in Georgia in early March. He had started on June 1st, but told me that he had just taken his time.
Later up the trail, I read a register far enough back to find his Christmas day entry, alone in a shelter. And I know it sounds crazy, but I had a little envy for him. It seems that that kind of solo would take a special kind of strength, and that it would be an incredible kind of solitude. And there’s a fantastic quiet when there’s a blanket of snow on the ground.
Tang