Surprised no one mentioned Moose Skowron. As a goofy little kid, I was thrilled when they picked him up, and later Rocky Colavito. The coolness of those names alone had to have added a year to their careers. They got more out of Moose than out of Rocky, but that's not saying much.
With the "not necessarily washed-up" caveat, Early Wynn and Ted Kluszewski meet the past-their-prime standard. They were before my time, and yes, I realize Wynn won the '59 Cy Young and wasn't bad in '60.
Smokey Burgess was no longer a full-time player but still pretty legendary by pinch hitting standards, such as they were.