I've put out one of my covered cat beds that I've wrapped in a fleece blanket and also put a fleece blanket inside. I then heat up a Snuggle Disc (a great safe device that you heat in the microwave and it stays very HOT for 6-8 hours, then it will slowly lose heat but stays nice and toasty warm for 10-12 hours) and put that under the fleece blanket. I've seen him curled up and sleeping inside it during the day. But, he always leaves at night, which makes me think he belongs to someone. Lots, and LOTS of folks in my neighborhood have pet doors.
I'm hoping he will continue to get used to me so that I can eventually pet him and pick him up. That's how Greybeard became part of my family. My neighbor and I have put out a couple of cat traps but so far he has not gone near them. It's also been too cold lately (I feel) to try and trap him at night.
I certainly don't need another cat, so I'm hoping I can figure out if he does really belong to someone. But, I can't turn my back on him either. So, for now, Pyewacket will continue to get fed and have a warm place to sleep at Possumlady's Home for Wayward Animals.