My mother had an UTI. It is cleared up, but she still has the urge to pee a lot, even right after she has just peed. Suggestions?

It is possible that her bladder is not fully emptying.
Try changing her position for 5-10 minutes before she gets up to go to the bathroom.
If she is in a recliner recline it a bit more. If she is sitting upright, recline her. If she is reclined or in bed sit her up for a bit.
Any of those changes might allow the bladder to empty a bit more

Agree with what Grandma1954 suggested. Also, UTIs are not only caused by poor hygiene, but by physiological changes in our bodies as we age, and other factors that are difficult to control. With my 85-yr old MIL in LTC we had success in cutting down the number of her infections by giving her a supplement called Mannose-D. It's a simple sugar and the science behind it is that it somehow helps them to empty their bladders. You can't overdose on it since it just gets peed out. Once your mom is cleared of UTI you can give it to her per the dosing recommendation on the label. I got mine on Amazon.

Are you sure the UTI is cleared, did they lab test a new urine sample?

Have you tried Kegel exercises?   Strengthening the supporting/surrounding muscles can help control the physical sensation of needing to urinate.