stray cat tail injury- skin chewed off

I found a stray cat a few weeks ago. his tail fur was matted and he yelped when I touched it. he had what appeared to be a scratch on his tail. I cleaned it multiple times a day because it looked like it was starting to get infected, it started looking better about a week later and then in a day he pulled off all the fur in the surrounding area. just when it looked like it was scabbing over and going to heal, when I came home from work that day- all the fur from the area was gone, the scab was gone and the top layer of skin was missing. it’s about a one inch by one inch patch near the base of his tail without skin. it doesn’t bleed. he eats and drinks and moves around. I don’t know if he has a fever or not, he doesn’t meow like he’s in pain, he generally sleeps and acts happy but it looks painful. I can see the shape of his tail. not bone, but he’s a few layers of skin away. I don’t know what to do. I’ve never had a cat before and I thought it would heal on it’s own. does he need a head cone to leave it aloneand a tail wrap with antibiotics or is this an ‘only a vet can fix it now’ injury? thank you for your time. I’m worried about him :-(