My cat's eye is swollen, please help!

Hi everyone. My cat’s left eye was swollen shut when he came back inside today however for the past few hours I used a cold, wet towel to help the swelling and used a saturated black tea bag to help with the swelling too, as well as to use as an antiseptic. Now he can open his eye almost fully, but there is still swelling around his left cheek underneath his eye, and his nasal canal looks wider than usual, though there doesn’t seem to be any swelling there. He is an outdoor cat (he only goes in our backyard) and has had no medical problems in the past, other than a cold a few months back. There are no obvious cuts or bruises around his face, but there was some dirt on his head, which I thought may lead to the conclusion that he was poked in the eye by some branches or twigs. There is no redness in his eyes. He is 2 years old. I was wondering whether I should let the swelling go down, wait and continue with the moist towels (is warm or cold better?) or whether this is something that requires immediate attention of a vet. Thanks! I’ve added pictures of what he normally looks like (the first one) versus now.