Various novel methods of forming a gate-to-source/drain conductive contact structure and the resulting novel device structures are disclosed. One illustrative method disclosed herein includes performing at least one first etching process to form a recess in a gate structure of a gate of a transistor device so as to expose an innermost surface of a portion of a sidewall spacer positioned adjacent a first sidewall of the gate structure and performing at least one second etching process through at least the recess in the gate structure so as to remove at least a portion of the portion of the sidewall spacer with the exposed innermost surface.