One illustrative method disclosed includes, among other things, forming a silicon dioxide etch stop layer on and in contact with a source/drain region and adjacent silicon nitride sidewall spacers positioned on two laterally spaced-apart transistors having silicon dioxide gate cap layers, performing a first etching process through an opening in a layer of insulating material to remove the silicon nitride material positioned above the source/drain region, performing a second etching process to remove a portion of the silicon dioxide etch stop layer and thereby expose a portion of the source/drain region, and forming a conductive self-aligned contact that is conductively coupled to the source/drain region.