One method disclosed herein includes, among other things, forming a patterned fin having a thickness that is equal to or greater than a target final fin height for a replacement fin, performing an etching process through the patterned fin etch mask to form a plurality of trenches in a semiconductor substrate to define a substrate fin and forming a recessed layer of insulating material in the trenches so as to expose the patterned fin etch. The method also includes forming a layer of CTE-matching material around the exposed patterned fin etch mask, removing the patterned fin etch mask to thereby define a replacement fin cavity and expose a surface of the substrate fin, forming the replacement fin on the substrate fin and in the replacement fin cavity, removing the layer of CTE-matching material and forming a gate structure around at least a portion of the replacement fin.