For Red River Gorge you have several great options.
Hanson's point. Doing the loop from the Grey's Arch Parking lot around to Hanson's point and Grey's arch is a great trip. It is outlined here.

Another is a little further down the road from Grey's arch, Auxier Ridge trail with Double Arch and Courthouse rock are interesting features with great views. The trip is outlined here.
As others have said a car hang tag is required but is only $3 for 1 night and $5 for up to 3 nights. No tag needed for day only visits only for overnight backwoods camping. Make sure you check the NF web site for closures. Auxier ridge could be closed if snow covered.

In Ohio do not miss The Hocking hills. This will be camp ground camping and day hikes, but the gorge, Cedar Falls, and Ash Cave makes a nice 12 mile round tip hike. Details here: Note: Their point 3 camping spot is illegal as dispersed camping in the park is not allowed. This area is too small and fragile to condone stealth camping.

Also there is Zaleski State Forest: The trip shown here, is a nice 1 night loop and it takes you to the old "Haunted" Moonville rail tunnel. Note: There are two potentially waist high creek crossings, at mile 9.8 ans 10 on the map. These can be avoided by following the creek bank along the north side for 1500'. Little bushwhacking but not bad in winter. Then again the creek may be low... I went in Spring.

Shawnee, Marietta, Wildcat Hollow are all also good. If going through Dayton area make sure to book a day for the National Museum of the US Air Force. 5 huge hangers of the history of military Aviation, including an SR-71, B-70, F188, WWI and WWII, etc. also included are the retired Presidential planes including the plane that flew Kennedy home from Dallas and where LBJ took the oath of office. Really cool stuff and all free. Contact TrailHeadz Hammocks also, as they are nearby in Bellbrook, OH.

Just some ideas.