Start with John's recommendation of the ACA routes, then spend an evening looking at weatherspark.com to check out various cities on the routes. Look at the averages, and pay attention to the 10% low temps.
I'd head east or south, aiming to get east of Charlottesville on the TransAm by the end of October. The idea would be to get out of the Appalachians pretty quick -- you can get snowstorms in the mountains by late October. Going south, you should miss cold weather fairly easily. You could head south on the Atlantic Coast from Virginia -- but stay off the Outer Banks in NC. Given six weeks, if you go south, you could go east on the Southern Tier, and perhaps south towards Key West. If you turn west on the ST, you'd have to push to get across the Rockies, or end your trip around El Paso.
Anywhere but on the gulf coast, I'd suggest you have enough financial reserve to spend a couple nights in a motel. You can get an early winter storm in November that'll give you some sleet or snow, but it'll probably melt after a day or two.