Call Exactly 30 days before you begin your trip. Start calling at 8 AM EST in the Morning !! The hardest shelter to get will be Ice Water Springs, however if you start at Newfoundland Gap, your first shelter will most likely be Pecks Corner, which shouldnt be too hard to get.
Remember as a section Hiker, you can only stay in he shelters you reserve, so make your picks carefully.
I would do them in this order:
Pecks Corner 10.9 miles
Tricorner 5.9 miles ( short day but good trip, you go over 2 mountains )
Cosby Knob 8.0 miles
Davenport Gap 9 miles ( includes the .6 mile side trip to Mt Cammerer Lookout, dont miss this one, its great )
The next morning get an early start,walk out of the park, turn right onto the GRavel road and hike 1.3 miles to Mountain Mommas. One Half pound cheesesburgers , you will kill for
Try to get a ride back to the trail head as early as possible ( Momma may be taking people there about 9:00 AM, so get to Mt Mommas early if possible ). Continue on Trail across Interstate and up Snow Mt. Stop at first Shelter after top of the Mountain.
PS: Momma only takes Cash !!!
This is a Great Hike, in Fact its one of my Favorites.
It takes 8 days at an easy pace or 6 to 7 hiking Fast. I think Sweeper did it in 6 because I missed him by one day, he was hiking so fast 
You may go to Cosby Knob on your 2nd Day instead of Pecks Corner, but this will be almost 14 miles and pulling lots of hills and mountains. If it rains, go shorter mileages.
You can go from Davenport Gap to Hot Springs in 3 and a half Days easy.
Have fun and I hope to see you on the Trail.
Jim Deane, Rebel with a Cause !!
Jim Deane