Since you haven't talked to the center manager about the situation I would try this approach:
The next time you must begin the day with a partial tank, go to the center manager before start time and calmly discuss the problem.
Key Points
I'm starting the day with a partial tank of fuel when it should be full.
In order to "make it work" I must fuel during my NDA delivery window which perpetuates the problem because my tank will also be low tomorrow. That diversion also puts my time commit pkgs. at risk of being late.
I don't have a credit card, and I only carry enough cash for lunch.
Will you issue me a fuel card so I can better perform my job?
(They like it when you present a problem and offer a viable solution in the same conversation.)
If he refuses to provide a fuel card, you should operate the truck until it is about dry, then park it and make the call.