One method disclosed herein includes forming an opening in a layer of material so as to expose the source/drain regions of a transistor and a first portion of a gate cap layer positioned above an active region, reducing the thickness of a portion of the gate cap layer positioned above the isolation region, defining separate initial source/drain contacts positioned on opposite sides of the gate structure, performing a common etching process sequence to define a gate contact opening that extends through the reduced-thickness portion of the gate cap layer and a plurality of separate source/drain contact openings in the layer of insulating material, and forming a conductive gate contact structure and conductive source/drain contact structures.