Glad the little new guy is getting better.
You likely already know that daily water changes will help him be his best in time to recover from the fin rot.
For help in recovering, I also like to use Epsom Salts - Magnesium sulfate (not really a salt).
You can start with plain Epsom Salts and mix 1 tsp per gallon to start. Dissolve in warm or hot water for the tank, treat with dechlorinator after making room /tank temp. add into tank 1/4 portion every ~8 hours. Then just make up a concentrate to use when you do a water change. Add that total of 1 tsp. per gallon.
I will use up to a tablespoon per gallon. Then if the Betta seems like he is doing well with the Epsom salts and fins still need help, I would increase the amount to the total of 1 TBL per gallon for 7-14 days.
If needed a bit longer, but then just reduce in water changing down to 1tsp. per gallon and keep 1 tsp total per gallon for another couple weeks.
Same way decrease the amount buy doing water changes as usual and see how he is.
Maybe in a month fins will be nicely growing back!
At any time along the way medicating may be warranted. Treat if needed, whatever is in his best interests.