The AT is a fairly unique situation, also. You have more shelter options than what you bring with you, so keep that in mind.
While you need to be prepared for any weather, you can also stop at a shelter and wait it out, or even stay in the shelter for the night. They're fairly well spaced out, so you can probably hit at least 1 or 2 a day, maybe 3 or 4 depending on your daily mileage, and the area you're in.
While I personally wouldn't go with a minimalist tarp, I also probably wouldn't take a large tarp with doors either. A WB Edge or Mamajamba would probably do for you. If you carry a poncho, you can also use that to supplement your tarp for coverage.