NEW YORK—Quinn Lemley, an actress and chanteuse, has specialized in portraying Rita Hayworth in a one-woman show—channeling the 1940s glamour queen's songs and mannerisms, sashaying across the stage in elegant evening gowns reminiscent of old Hollywood.

But on some mornings after a performance, Ms. Lemley would report to Weill Cornell Medical College for a very different kind of acting job. Her costume for these gigs is two hospital gowns, one wrapped artfully on top of the...