It's not that you're rushing her. She's probably so busy enjoying her walk that she just doesn't take the time to completely empty her bladder. In a puppy mill situation there is seldom a need to learn to empty the bladder completely as they go right in their cages. You're fortunate she doesn't soil her crate.

Peeing often and in small amounts is usually the first and most obvious sign of a UTI too.

Good luck, I hope it goes well and she just has to learn to relieve herself completely before going back inside.