An illustrative example elevator assembly includes a header beam and a first elevator cab supported by the header beam. A plurality of vertically oriented rods extend beneath the first elevator cab. A horizontally oriented mid-beam is coupled to a first one of the rods near a first end of the mid-beam and coupled to a second one of the rods near a second end of the mid-beam. A second elevator cab is situated beneath the first elevator cab and beneath the mid-beam. At least one linear actuator is supported at least partially on the mid-beam. The linear actuator selectively causes vertical movement of the second elevator cab relative to the rods.