Methods of forming a gate-to-source/drain contact structure

One illustrative method disclosed herein includes, among other things, performing at least one etching process to expose at least a portion of an upper surface of a gate electrode of a first transistor device and at least a vertical portion of one side surface of the gate electrode and performing a material growth process to form a conductive gate-to-source/drain (GSD) contact structure that conductively couples the gate electrode of the first transistor device to a source/drain region of the first transistor device, wherein the conductive GSD contact structure comprises a non-single crystal material portion positioned on previously exposed portions of the gate electrode and a single crystal material portion positioned in the source/drain region.