Remember, in the Third Age Aman and Eressea were no longer part of "the Earth;" Tolkien implies at the end of Akallabeth that the proto-Americas were created at the time of the Downfall and the World Made Round. I have no doubt that ultimately Sauron would have liked to control the New (literally) World as well.
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The entire plot of The Lord of the Rings could be said to turn on what Sauron didn’t know, and when he didn’t know it.
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