One method disclosed herein includes forming a sacrificial etch stop material in a recess above a replacement gate structure, with the sacrificial etch stop material in position, forming a self-aligned contact that is conductively coupled to the source/drain region, after forming the self-aligned contact, performing at least one process operation to expose and remove the sacrificial etch stop material in the recess so as to thereby re-expose the recess, and forming a third layer of insulating material in at least the re-exposed recess.