My T doesn't charge for emails or calls either.
One thing I want to mention is that I hope you aren't going to be hurt if for some reason T does not address the extra payment or say you don't need to pay. She might just accept it without comment, she may not even notice actually (she might have an office manager that would taka care of the deposit,etc ) or any one of any number of different scenarios. I just hope that you weren't hoping that she would use this as an opportunity to reassure you of her care.