One method disclosed herein includes forming an etch stop layer above recessed sidewall spacers and a recessed replacement gate structure and, with the etch stop layer in position, forming a self-aligned contact that is conductively coupled to the source/drain region after forming the self-aligned contact. A device disclosed herein includes an etch stop layer that is positioned above a recessed replacement gate structure and recessed sidewall spacers, wherein the etch stop layer defines an etch stop recess that contains a layer of insulating material positioned therein. The device further includes a self-aligned contact.