A series of hafnia/silica, oblique incidence (56 degree(s)), 1064 nm high reflectors (HRs) were prepared and coated with silica overlayers of varying optical thickness from (lambda) /2 to 4(lambda) in order to determine the effect of an overlayer on the laser-damage resistance of the HRs. The stress and laser damage thresholds for S and P polarization of the HRs were measured, and the damage sites for P polarization examined by Atomic Force Microscopy. All the multilayers were found to be in compression, with an intrinsic stress increasing with overlayer thickness. The presence of an overlayer and its thickness did not affect the damage threshold significantly. However, the presence of an overlayer greatly influenced the size and morphology of the damage. First, the overlayer prevented catastrophic `burns' of the hafnia top layer. Second, as the overlayer thickness increased, two distinct damage morphologies were found: jagged pits and round craters. The diameter of these pits and craters then increased somewhat with thicker overlayers. The depths of the pits and craters also increased with overlayer thickness, and the depths showed failure occurring at the interfaces below the hafnia layers. The side-wall angles of the craters were shallower with thicker overlayers, but there was no angle dependence for the pits. The craters showed fracture-like features and a small hillock or pit on their bottom surfaces. No correlation of damage morphology to conditioning or fluence was found.