One illustrative method disclosed herein includes, among other things, forming a layer of insulating material in the source/drain regions of the device, wherein the layer of insulating material has an upper surface that is substantially planar with an upper surface of a gate cap layer, recessing the layer of insulating material such that its recessed upper surface exposes a surface of the fin, performing another etching process to remove at least a portion of the fin and thereby define a recessed fin trench positioned above the recessed fin, and forming an epitaxial semiconductor material that is at least partially positioned in the recessed fin trench.