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From that time war never ceased between Sauron and the Elves... But Sauron gathered into his hands all the remaining Rings of Power; and he dealt them out to the peoples of Middle-earth, hoping thus to bring under his sway all those that desired secret power beyond the measure of their kind. ("Of the Rings of Power and the Third Age")
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My mistake, Earendilyon. This passage shows explicitly that Sauron gave the Nine Rings and Seven Rings to Men and Dwarves, respectively,
after he was revealed by Celebrimbor. I suppose then, that it is entirely possible that his Rings would still have been in demand to some. However, he could not hope to attain much more power than he already had since he had probably already dealt out Rings to all the powerful leaders who desired them. Any attempt to give Rings of Power to anyone else of import would probably have proven folly, and thus it would have been a pointless exercise to creat more Rings.