Concerned about REU eligibility given new graduation plans
Some research experience for undergraduate (REU) programs are available to students who have recently graduated. Therefore, while programs are by no means as plentiful as for enrolled undergraduates, they are still out there, and you can avail yourself of the opportunity to participate in one of those programs.

However, the important point is that you show participation in research; the mechanism is usually of lesser importance. The REU just might be the most convenient way to do that in mathematics.