One method disclosed herein includes, among other things, a method of forming a contact structure to a source/drain region of a transistor device. The transistor device includes a gate structure and a gate cap layer positioned above the gate structure. The method includes forming an extended-height epi contact structure that is conductively coupled to the source/drain region. The extended-height epi contact structure includes an upper surface that is positioned at a height level that is above a height level of an upper surface of the gate cap layer. The method further includes performing an etching process to trim at least a lateral width of a portion of the extended-height epi contact structure, and, after performing the etching process, forming a metal silicide material on at least a portion of the trimmed extended-height epi contact structure and forming a conductive contact on the metal silicide material.