One illustrative method disclosed herein involves forming an etch stop layer above a plurality of sacrificial gate structures, performing an angled ion implant process to implant an etch-inhibiting species into less than an entirety of the etch stop layer, and forming a layer of insulating material above the etch stop layer. The method further includes removing the sacrificial gate structures, forming replacement gate structures, forming a hard mask layer above the replacement gate structures and layer of insulating material, forming a patterned hard mask layer, performing another etching process through the patterned hard mask layer to define an opening in the layer of insulating material to expose a portion of the etch stop layer, performing another etching process on the exposed portion to define a contact opening therethrough that exposes a doped region and forming a conductive contact in the opening that is conductively coupled to the doped region.