One illustrative transistor device disclosed herein includes a final gate structure that includes a gate insulation layer comprising a high-k material and a conductive gate, wherein the gate structure has an axial length in a direction that corresponds to a gate width direction of the transistor device. The device also includes a sidewall spacer contacting opposing lateral sidewalls of the final gate structure and a pillar structure (comprised of a pillar material) positioned above at least a portion of the final gate structure, wherein, when the pillar structure is viewed in a cross-section taken through the pillar structure in a direction that corresponds to the gate width direction of the transistor device, the pillar structure comprises an outer perimeter and wherein a layer of the high-k material is positioned around the entire outer perimeter of the pillar material.