Round and Round the Mulberry Bush with half a pint of moonshine: Doyle, Debra and James D. Macdonald. Timecrime Inc. Edited by Robert Silverberg, Tim
Dave, Jane, Bruce and Cynthia each receive a visit from a man who tells them not to join the Time Patrol or else.
Worried, they track him (via his clothing and accent) to 1920s Chicago where they discover that Al Capone has acquired a time belt and is using it to "work" the future. They leave their time tour and themselves become timecrimers. As part of their guise, Dave becomes a gangster ("white ribbon Dave" because he tells people he's signed the pledge) while Jane becomes his ultra-competent moll. Bruce gives advice, and Cynthia rather disappears from the story.
What follows is the most fascinating chase through time, creating time loops and, towards the end, cleverly explaining how those loops are closed. Frankly, I'm not good enough to explain it all to you, but for a brief moment at the end of the book I understood how time paradoxes work and how you can close them.