First of all, go ahead and remove the filter completely. Since this is a hospital tank and I assume you're treating your betta for something, you're going to be doing frequent 100% water changes and not cycling(which a 1 gallon is too small for anyway)so the filter is useless, and in such a small space will probably do nothing but push your fish around with the current.
Next, completely dissolve 1 tsp. of AQ salt in a small, clean fish-safe(no soap residue or other chemicals on it)cup or container with a little hot water and pour it into the tank.
Then, acclimate your fish into the hospital tank like when you first brought him home. Scoop him up in his petstore cup, and assuming the hospital tank is within a couple degrees of his normal home, pour out a little of his old water and add a little of the new water every 10-15 minutes a few times, then gently release him into the tank.
What are you treating him for?