I don’t know how far north you plan to travel. There are several short hikes from Skyline Drive in Shenandoah National park plus a drive up or two.
Way down south though not near the Blue Ridge Parkway one to consider would be White Top, W4V/WV-002. It’s a drive up worth 10 points and the view is fantastic.
I’d say check with some of the W4V/W4C activators. I’m sure they all have some good first hand knowledge.