(Drift Woody @ Mar. 19 2014, 6:43 am)
Sheltowee Trace is a linear trail.
Red River Gorge is very rugged (and very scenic) and loops are possible, but it's also very popular and therefore not very remote (if your definition of "remote" is getting away from people). The Clifty Wilderness section is perhaps not as heavily used, but some road walking would be needed for a loop.
Big South Fork National River and Recreation Area is south of RRG, stretching from Kentucky into Tennessee. There are many long loop possibilities there.
Great advice, I'll add a few things though . . .
The Sheltowee Trace is a linear trail, but you can combine it with other trails to make a loop in certain places (the 11-mile partial loop that starts at the Hwy. 192 ST parking area that goes by Van Hook Falls, then follows Cane Creek to the Rockcastle River and loops back to the falls is probably my favorite). There are also a few shuttle services than can drop you off and then you hike back to your vehicle, which makes doing a section hike of the ST possible.
I would avoid the RRG on weekends if you're looking for solitude. During the week you could have miles of trail and campsites to yourself, and there are several nice loops possible. I'd recommend picking up the book "Hinterlands" that details the unmarked trails, because when combining the unmarked trails with the official trails you can hike some great routes.
Big South Fork NRRA is probably your best bet for solitude, big loops and a remote feeling. Amazing area . . . one of the most incredible parcels of public land on the Cumberland Plateau without a doubt.
Also, you might want to check out www.kywilderness.com and ask on their forum, as well as browse past trip reports from backpackers in the state.
Happy hiking, and I'd be happy to answer any other questions you have. I've done most of my backpacking in Kentucky and I'd be happy to give you more details once you focus in on where you want to go.
Mostly Southeastern Outdoor Scenery: http://www.flickr.com/photos/mvweth