A recent ferromagnetic resonance study [Jeon et al., Nat. Mater. 17, 499 (2018)] has reported that spin pumping into a singlet superconductor (Nb) can be greatly enhanced over the normal state when the Nb is coupled to a large spin-orbit-coupling (SOC) spin sink such as Pt. This behavior has been explained in terms of the generation of spin-polarized triplet supercurrents via SOC at the Nb/Pt interface, acting in conjunction with a nonlocally induced magnetic exchange field. Here we report the effect of adding a ferromagnet (Fe) to act as an internal source of an additional exchange field to the adjacent Pt spin sink. This dramatically enhances the spin pumping efficiency in the superconducting state compared with either Pt and Fe separately, demonstrating the critical role of the exchange field in generating superconducting spin currents in the Nb.