Check the fine print with your loan provider, but many charge additional fees if you pay via credit card. Mine will charge a flat fee of $14.95 for a credit card payment, but I've also seen 2.5-3.5% of the payment. Card churning with student loans is probably not a good idea.
However, you could look into transferring the balance onto a 0% credit card and paying off the balance without interest. Given your total is around 60k, you might have to transfer multiple times. Run the numbers though, you still might save in the long run compared to paying 6.55% interest. I personally made a topic on it a few months ago. Maybe that could help you!