My advice for an AT section is a simple cannister stove like the MSR Pocket Rocket or Snow Peak Gigapower. Cannisters are easy to resupply in places like Erwin or Hot Springs or Gatlinburg or at the Hike Inn or Fontana Village on the southwest side of the Park or Nantahala Outdoor Center (NOC) or Franklin, NC. This pretty well covers the entire NC corridor.

I recommend a cannister stove despite the somewhat extra weight because of the flexibility it provides. You can buy lipton dinners or mac 'n' cheese at any grocery store en route, which require a decent bit of boil time, something esbit tabs or alcohol stoves are not so great with. If you send maildrops to yourself with simple add-boiling-water meals, you may find alcohol or (my personal lightweight favorite) esbit tabs work fine, are very lightweight and relatively cheap compared to buying a $30-40 stove plus cannisters.