Enhancement of Gilbert damping in polycrystalline cobalt thin-film multilayers of various thicknesses, overlayered with copper or iridium, was studied in order to understand the role of local interface structure in spin pumping. X-ray diffraction indicates that cobalt films less than 6 nm thick have strong fcc(111) texture while thicker films are dominated by hcp(0001) structure. The intrinsic damping for cobalt thicknesses above 6 nm is weakly dependent on cobalt thickness for both overlayer materials, and below 6 nm the iridium overlayers show higher damping enhancement compared to copper overlayers, as expected due to spin pumping. The interfacial spin mixing conductance is significantly enhanced in structures where both cobalt and iridium have fcc(111) structure in comparison to those where the cobalt layer has subtly different hcp(0001) texture at the interface.