In superconducting Co∕Pt–Nb–Co∕Pt multilayers with perpendicular magnetic anisotropy, the superconducting transition temperature Tc is considerably higher in the parallel (P) orientation of the two Co∕Pt magnetizations compared to the antiparallel (AP) orientation. Low temperature magnetic-force microscopy reveals that the individual ferromagnetic layers are uniformly magnetized in both configurations and do not exhibit a magnetic stray field arising from a multidomain state. We also demonstrate that the difference ΔTc=TcP−TcAP between both Tc values is enhanced at higher transport currents due to the increasing imbalance of spin-polarized charge carriers in the superconducting Nb layer. Hence, the spin switch can be controlled by the transport current in future superconducting devices.