One illustrative method disclosed herein includes, among other things, forming first and second vertically-oriented channel (VOC) semiconductor structures for, respectively, first and second vertical transistor devices, and forming first and second top spacers, respectively, around the first and second VOC structures, wherein the first spacer thickness is greater than the second spacer thickness. In this example, the method also includes performing at least one epitaxial deposition process to form a first top source/drain structure around the first VOC structure and above the first top spacer and a second top source/drain structure around the second VOC structure and above the second top spacer, and performing an anneal process so as to cause dopants in the first and second doped top source/drain structures to migrate into, respectively, the first and second VOC structures.