This is a pursuit that I have dedicated considerable research time to, over the years. My single best find is a fair drive from Ktown, but certainly takes on the flavour of a nice 'picnic adventure'. Are you familiar with Prince Edward County? Google Map the following Conservation Area - and I am not joking about what it's called: Beaver Meadow Wildlife Management Area. I haven't been there for about 5 years, so have no idea what its current presentation is like. When I was there - with my friend... - it was decidedly off the grid. Quite a long, unpaved access road ends in a parking area. It's nicely laid out, meaning that the parking is a fairly hard left hand turn at the end of the road. If you are in the lot, you are well out of sight of anything and everything, and should probably be able to hear any traffic approaching.
The Area itself has always seemed quite odd to me. I think at one time it was used for duck hunting - we found a number of slightly delapidated 'look outs' that I think might have functioned as blinds at one time. Pressure treated lumber - about 8 x 12 or so, with a kind of 'railing+bench seating' around the perimeter. It took a bit of work to find them. Back then they were very grown over. No idea if they are now totally grown over or may have been spiffed right back up again. You really would have to go check it out to see what I mean.
And this time of the year? I'd stick with the parking lot, or bathe yourself in bug spray first. Have fun.