Most cases of fin rot can be healed with clean, warm water along with good nutrition.
Since your fish is in a 0.5 gal tank, I recommend doing daily water changes. This will keep the levels of toxins, wastes and bacteria low. This lessens the chance for infection, and allows the fins to heal.
Also, bettas are tropical fish and do best at temperatures between 76-84F. For most of this, this means using a heater (along with the thermometer) for at least part of the year (if not the full year).
How much Aquarium salt have you been using? And for how long?
Aquarium salt can be used SHORT term. It contains sodium, which causes fluid retention and puts stress on the kidneys when used in high dosages. Therefore, I would limit use of this to situations where there is an active external infection.
If the rot continues to worsen or gets close to his body, that would be the time to consider antibiotics.
Can you post a photo of him?