One illustrative method disclosed herein includes a forming plurality of trenches in a substrate to thereby define a fin structure for a FinFET device, forming a first region of a first insulating material within each of the trenches, wherein the as-deposited surface of the first insulating material is positioned below an upper surface of the fin, forming a layer of a second material that contacts the as-deposited surface of the first region of the first insulating material and overfills the trenches, performing at least one process operation to remove at least a portion of the layer of the second material from above the fin structure, and, after performing the at least one process operation, performing a second process operation to selectively remove the second material from above the first region of the first insulating material and thereby expose the as-deposited surface of the first region of the first insulating material.