Ooh, thanks to all for the information and ideas. I had issues with both opossum and raccoons at my old place, but here, less than 1000 feet away, I've only seen opossum. Mr. Casper, my stray, had a little house on the covered deck at my old place, and he has two here in the screened-in porch. I've seen young opossum come into the screened-in porch for food, but as far as I know, there have never been any issues about shelter.
I have heard that opossum will avoid light. Does anyone know if that's true? If so, you could try leaving an outdoor light on. I am thinking of having a motion-sensor light installed at the rear of the porch, where there's an opening for Mr. Casper (and other critters) to get in and out.