Well, shoot. Now what? I went in to tell Eric and ask if he though my idea might be worth trying. So we got one of the cleaned out, old recycling cans from the garage and a smooth dish towel to make a little "nest" with. Then we scooped up the soggy little peeper and placed it inside.
At first we set it on the picnic table, but since there was also a relatively dry white breasted nuthatch cheeping and hopping about nearby my truck we moved it over to some brush nearby - on top of that dumb old rototiller that the last people left as decoration and we have yet to remove. I ended up rolling up another towel and making a U-shaped bolster of sorts around the back to hopefully trap some of it's body heat to help it warm up and dry off. There's no photo of that.
Not too awfully long after, certainly not longer than 30 minutes, Eric went out to check on it then came up to tell me, "It's like a totally different bird. All fluffy and peeping." So came right down to check and it was already hopping around on a rock outside the ad hoc shelter! And by the time I got my camera activated it was hopping up the nearest tree! Wanting nothing to do with me, of course.
Turns out it pooped on my towel, too.