Gate contact structure for a transistor device

One illustrative transistor device disclosed herein includes, among other things, a gate positioned above a semiconductor substrate, the gate comprising a gate structure, a conductive source/drain metallization structure positioned adjacent the gate, the conductive source/drain metallization structure having a front face, and an insulating spacer that is positioned on and in contact with at least a portion of the front face of the conductive source/drain metallization structure. In this example, the device also includes a gate contact opening that exposes at least a portion of the insulating spacer and a portion of an upper surface of the gate structure and a conductive gate contact structure positioned in the gate contact opening, wherein the conductive gate contact structure contacts at least a portion of the insulating spacer and wherein the conductive gate contact structure is conductively coupled to the gate structure.