Three out of the last four years I've ridden my bike trip SS, okay so last year was the only truly SS the rest of the time I just didn't change gears. I was riding upper midwest, mid-Atlantic, and New England, where you do have plenty of hills and I had not problems riding SS. I was going to go FG this summer but after getting an offer I couldn't refuse I ended up buying a new and went multi-speed this year. Once I got down south I didn't change gears much though. I'm not sure if it was the weather or the packing method, which also changed this year, but I was glad, especially until I got into southern SC that I had went multi-speed instead of fixie this year. I'm not what was going on early in the trip. Everything this year was radically different compared to the previous three years so I have no real way of comparing anything to nail down what was causing what I was seeing happen this summer.
I would think the only area where you would see any real trouble would be out west when you hit the climbs...not sure what route the Southern Tier follows but I don't think you can avoid the climbs anywhere. If your in good shape now you'll be in even better shape by the time you get there though and that should help quite a bit.