I believe it's true to say, is it not, that "Middle-earth" as a continent included the vast lands of Rhūn and Harad as well as "the West", ie Gondor, Rohan, Rhovanion and Eriador? At the end of the Third Age Sauron already more or less controlled all of the East and the South which seemingly made up the majority of Middle-earth's landmass.
It might be worth noting this remark from letter 183:
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Sauron desired to be a God-King, and was held to be this by his servants; if he had been victorious he would have demanded divine honour from all rational creatures and absolute temporal power over the whole world.
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As well as this remark about Sauron from
Morgoth's Ring:
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He probably deluded himself with the notion that the Valar (including Melkor) having failed, Eru had simply abandoned Eä, or at any rate Arda, and would not concern himself with it any more. It would appear that he interpreted the "change of the world" at the Downfall of Nśmenor, when Aman was removed from the physical world, in this sense: Valar (and Elves) were removed from effective control, and Men under God's curse and wrath.
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What this would suggest to me is that Sauron's policy in the Third Age was the domination of all of Middle-earth but just Middle-earth; that way he would be the master of as much of Arda as possible while still apparently being free of the interference of the Valar or Eru (although of course he was lying to himself when he thought that either party had abandoned Middle-earth). Sauron knew, or at least thought he knew, his limits, it would seem, and the most powerful he could be was as ruler of the mortal lands of Middle-earth.
So I would say the answer is this: Sauron did only want Middle-earth, but Middle-earth was more than just the West. That being said, he already controlled the rest of it. Only the West was holding out.