Cat shelters with capacity?

Not sure where else to post this but there’s this cat near my gym and he looks really skinny and underfed. I’ve left food out for him the past few days which he seems to eat.

The temperature is getting in the 30s again tonight and I’m worried about him being in the cold. I’ve tried approaching him so that I can grab him and take him somewhere warm but he’s really scared.

If I do capture him I’ll keep him for a few days but I’m definitely not in a place to be able to take care of a cat right now. I’ve called animal shelters and they invariably seem to be out of capacity and aren’t taking any new cats. Do y’all know of any smaller or unlisted shelters that I might be able to reach out to?