On my first cross country tour back in 1987, I started from Astoria on September 1 and got to the Outer Banks of NC in early November. I didn't do the Trans Am - but approximated it. I would say that it got a little bit iffy after October - the days just got so short. Halloween was in Blacksburg, VA - I remember. I had glorious weather across the West, snow in Lincoln, Nebraska, and quite a few yucky days in late October. I would say you could leave as late as August 15 for a planned 10-week tour. Then allow yourself a day here and there later in the tour for bad weather.
Fall touring is fabulous - just remember the days are short and bring something to read in your tent.