We have deposited polycrystalline Re-doped (Fe65Co35)((100-x))Re-x (0 <= x <= 12.6 at. %) thin films grown under identical conditions and sandwiched between thin layers of Ru in order to study the phenomenon of spin pumping as a function of Re concentration. In-plane and out-of-plane ferromagnetic resonance spectroscopy results show an enhancement of the Gilbert damping with an increase in Re doping. We find 98% enhancement in the real part of effective spin mixing conductance [Re(g(eff)(up arrow down arrow))] with Re doping. Conversely, the Re(g(eff)(up arrow down arrow)) does not change with Re doping in Fe65Co35 thin films which are seeded and capped with Cu layers. The enhancement in Re(g(eff)(up arrow down arrow)) of Re-doped Fe65Co35 thin films sandwiched between thin layers of Ru is linked to the Re doping-induced change of the interface electronic structure in the nonmagnetic Ru layer. The saturation magnetization decreases 35% with increasing Re doping up to 12.6 at. %. This study opens a direction of tuning the spin mixing conductance in magnetic heterostructures by doping of the ferromagnetic layer, which is essential for the realization of energyefficient operation of spintronic devices.