One illustrative method disclosed herein includes, among other things, forming a first gate structure above a fin, forming epi semiconductor material on the fin, performing at least one first etching process through a patterned sacrificial layer of material to remove at least a gate cap layer and sacrificial gate materials of the first gate structure so as to define a first isolation cavity that exposes the fin while leaving the second gate structure intact, performing at least one second etching process through the first isolation cavity to remove at least a portion of a vertical height of the fin and thereby form a first isolation trench, removing the patterned sacrificial layer of material, and forming a layer of insulating material above the epi semiconductor material and in the first isolation trench and in the first isolation cavity.