One illustrative method disclosed herein includes, among other things, forming a source/drain contact structure between two spaced-apart transistor gate structures, recessing the source/drain contact structure to define a source/drain contact etch cavity and depositing a conformal second layer of insulating material above a first layer of insulating material and in the source/drain contact etch cavity. The method also includes forming a third layer of insulating material above the conformal second layer of insulating material, forming an opening in the conformal second layer of insulating material and forming a V0 via that is conductively coupled to the exposed portion of the recessed source/drain contact structure.