Go here and put in the state you are from. IF you have a wyoming carry permit, I believe that you could concealed carry through much of the Appalachian trail.
http://www.usacarry.com/
Open carry would be determined by the individual states laws determining whether or not open carry is allowed.
Georgia has open carry and reciprocity with wyoming if you have your permit. With a wyoming permit this should mean you can concealed carry through SC. NC is an open carry state, TN is not an open carry state.
SC does not have reciprocity with Georgia carry permits.
Also a for a significant portion of the AT, it is illegal to discharge a firearm.