We demonstrate the strong tunability of the spin-pumping contribution to magnetic damping in a thin-film ferromagnetic free layer interfacing with a synthetic ferrimagnet (SFM), acting as a spin sink, via a thin Cu spacer. The effect strongly depends on the magnetic state of the SFM, a trilayer structure composed of two Fe layers coupled via indirect exchange mediated by a Cr spacer. With increasing Cr thickness, the SFM state undergoes a transition from an antiparallel configuration via a noncollinear configuration to a parallel configuration. We can explain the corresponding nonmonotonous dependence of spin relaxation in the free layer in terms of modulation of the longitudinal spin transport as well as relaxation of the transverse angular momentum in the SFM. The results should be useful for designing high-speed spintronic devices where tunability of spin relaxation is advantageous.