If you fly/train/bus into DC, you can start there or take a regional bus out to Cape Lookout - state park w/ camping.
From DC you can ride the C&O/Allegheny Crossing Trail to Pitt then take the Old Lincoln Highway across Ohio.
(sectional maps available at the Lincoln Highway Ohio Chapter website)
Northern Tier from Monroeville, IN (great cycling town) to Muscatine, IA
From Muscatine you would ahve to head out across Iowa and Nebraska before rejoining the TransAm.
From Jackson, Wyo you can head directly west to Oregon via Idaho Falls, Salmon, and Lowman.
(The Sawtooths are lovely in late summer.)
Westbound in Oregon is better on the TA than the Columbia Gorge because of winds.
(Years ago I could hardly pedal in my granny gear heading west in the Gorge.)