Something similar happened to my betta, the rotted pieces sort of fell off and then he was fine. I did extra water changes for a while, but he seemed to heal rather quickly. In terms of salt, you can add about 1 tablespoon of non-iodized salt per gallon (aquarium salt (not for saltwater tanks) is fine, but more expensive than what you can find in the grocery store). I'd guess you are in the ~1g range if you're using a vase, but you could go with 2/3 - 3/4 tablespoon instead. But if he seems to be getting better with no "meds" you might not need to go the salt route
Good luck!