V.N.
I would see if I could bring it to a local rescue group or animal control. Depending where you live some animal controls are better than others. Some will euthanize the animal after the holding period and some will try to find a shelter to take it.
As far as being a neighbor's cat if they are letting it out in these crazy temperatures perhaps the cat will be better off in a different home. I am a firm believer that cats should not be outside as they can become prey or injured. And of course any shelter/animal control will scan the cat first and animal control will fine the pet for being outside if it is returned.
Thank you for caring for this cat.