One method disclosed includes, among other things, forming a structure comprised of an island of a first insulating material positioned between the gate structures above the source/drain region and under a masking layer feature of a patterned masking layer, forming a liner layer that contacts the island of insulating material and the masking layer feature, selectively removing the masking layer feature to thereby form an initial opening that is defined by the liner layer, performing at least one isotropic etching process through the initial opening to remove the island of first insulating material and thereby define a contact opening that exposes the source/drain region, and forming a conductive contact structure in the contact opening that is conductively coupled to the source/drain region.