When I go for short overnighters, I like to pick a loop hike in the Shenandoah National Park. There's quite a few places that you can camp off the trail from the loop. They all feed off of the AT. You need to get a free permit at the ranger's station and if you have a loop in mind, ask them if they know of any spots. You can find the small pocket-sized book at any local outfitter store. I generally park further from the recommended start point, hike the AT and then pick up the loop. You can by section maps of the park as well
to plot your course of action.

Another choice is Cold Mountian/Mt Pleasants down in Buena Vista. My favorite way is to hike on the Hotel trail counterclockwise for around 4-5 miles,(across the balds) camp in this wonderful site...meaning it is flat, soft, and good views. Pack up the next day, or leave that site, and keep hiking about a mile which will bring you back to your car. Then I dump my gear, grab a day pack and hike up Mt Pleasants.(right by the car. No need to drive) That is a 6 miler up and back. There's great spots there to camp as well. The top of Mt Pleasant has wonderful views. I just did that over the holidays. Total miles for both days around 13. I can post some photos if I can figure how.

Also, just follow the AT for your chosen distance.

The SNP is going to be celebrating their 100th anniversary (or some such number, not sure what )so keep in mind it will be a busy spring and summer with lots of people.

Ramsey's Draft is another good choice. It's a Wilderness, so the trails aren't exactly well-kept, but it's a fun trail, good camping at Hiner Springs.