Competition between superconducting and ferromagnetic ordering at interfaces between ferromagnets (F) and superconductors (S) gives rise to several proximity effects such as odd-triplet superconductivity and spin-polarized supercurrents. A prominent example of an S/F proximity effect is the spin switch effect (SSE) observed in S/F/N/F superconducting spin-valve multilayers, in which the superconducting transition temperature Tc is controlled by the angle φ between the magnetic moments of the F layers separated by a nonmagnetic metallic spacer N. Here we present an experimental study of SSE in Nb/Co/Cu/Co/CoOx nanowires measured as a function of bias current flowing in the plane of the layers. These measurements reveal an unexpected dependence of Tc(φ) on the bias current: Tc(π)-Tc(0) changes sign with increasing current bias. We attribute the origin of this bias dependence of the SSE to a spin Hall current flowing perpendicular to the plane of the multilayer, which suppresses Tc of the multilayer. The bias dependence of SSE can be important for hybrid F/S devices such as those used in cryogenic memory for superconducting computers as device dimensions are scaled down to the nanometer length scale.