One illustrative method disclosed includes, among other things, forming a gate around an initial fin structure and above a layer of insulating material, and performing a fin trimming process on an exposed portion of the initial fin structure in the source/drain region so as to produce a reduced-size fin portion positioned above a surface of a layer of insulating material in the source/drain region of the device, wherein the the reduced-size fin portion has a second size that is less than the first size. In this example, the method also includes forming a conformal epi semiconductor material on the reduced-size fin portion and forming a conductive source/drain contact structure that is conductively coupled to and wrapped around the conformal epi semiconductor material