if he is eating and active i would recommend try stress coat with frequent water changes first. If you can't find stress coat just water changes. If he will not improve then aquarium salt.
Now i would recommend put him in smaller container or lower the water in your tank and do 100% water changes every other day and if not improvement then aq salt. About 100% water changes i know some people sceptical i do it all the time for my fish i don't cycle i just do 100% water changes. I am going to post two links for you so you can see check it out. First link you will see skinny sick malnutritioned betta that became a beauty with 100% water changes and second link some one treated betta with aq salt and daily water changes and stress coat
See page 1 page 43#422 page 42 and page 52 it all the same fish.
Also are you cycling your 6 gall. Do you have a filter? Make sure you swish/wash it regularly and vacuum the gravel regularly. And i would say if your fish got fin rot then something is wrong and i would increase water changes for life or just do 50% and 100% water changes .
Also with aquarium salt you really need to do 10 days. Aquarium salt dosage 1tsp/gall up to 3 tsp/gall with daily 100% water changes. Make sure you dissolve it first DO NOT ADD it directly to the tank.
When you change 100% make sure you acclimate your betta by adding a little bit of new water to his changing cup with original water and do it for a few 2-3,4 times and let him get used to 5 min each time when you add new water.