One method includes forming first sidewall spacers adjacent opposite sides of a sacrificial gate structure and a gate cap layer, removing the gate cap layer and a portion of the first sidewall spacers to define reduced-height first sidewall spacers, forming second sidewall spacers, removing the sacrificial gate structure to thereby define a gate cavity, whereby a portion of the gate cavity is laterally defined by the second sidewall spacers, and forming a replacement gate structure in the gate cavity, wherein at least a first portion of the replacement gate structure is positioned between the second sidewall spacers. A device includes a gate structure positioned above the substrate between first and second spaced-apart portions of a layer of insulating material and a plurality of first sidewall spacers, each of which are positioned between the gate structure and on one of the first and second portions of the layer of insulating material.