I looked up "westernmost island in Alaska", and found out that it is called Attu Island. However, this island is in the Eastern Hemisphere and can therefore also be considered one of the easternmost points in the US and even the world.
However, the geometrically easternmost island is Semisopochnoi Island, near the 180th meridian. I wonder which island lies beyond it, and can claim the spot of westernmost island geometrically and be relatively close to that in terms of the International Date Line sense as well.