One method disclosed herein includes forming a conformal liner layer in a plurality of trenches that define a fin, forming a layer of insulating material above the liner layer, exposing portions of the liner layer, removing portions of the liner layer so as to result in a generally U-shaped liner positioned at a bottom of each of the trenches, performing at least one third etching process on the layer of insulating material, wherein at least a portion of the layer of insulating material is positioned within a cavity of the U-shaped liner layer, and forming a gate structure around the fin. A FinFET device disclosed herein includes a plurality of trenches that define a fin, a local isolation that includes a generally U-shaped liner that defines, in part, a cavity and a layer of insulating material positioned within the cavity, and a gate structure positioned around the fin.