One illustrative IC product disclosed herein includes a gate structure for a transistor, a conductive source/drain contact structure and an insulating source/drain cap structure positioned above the conductive source/drain contact structure, wherein the insulating source/drain cap structure has a first notch formed therein. In one illustrative example, the product also includes a sidewall spacer that has a second notch in an upper portion of the sidewall spacer, wherein a first portion of the insulating source/drain cap structure is positioned in the second notch, and a conductive gate contact structure comprising first and second portions, the first portion of the conductive gate contact structure being positioned in the first notch and the second portion of the conductive gate contact structure being in contact with the gate structure.