So the answer was found in the comments. And the fact that I misworded the question was there too. Oh well. Anyway:
Who was the last incumbent President to fail to win reelection in a two-way race? (corrected question)
It was Herbert Hoover. Yes, that’s right, incumbents seeking reelection have won every time they’ve tried for 80 years. In 1992, H. Ross Perot captured 18.9% of the vote when George H.W. Bush lost. In 1980, John Anderson garnered 6.6% as Jimmy Carter went down to defeat. Gerald Ford was never elected in the first place so it wasn’t a reelection. And that’s it. An impressive trend and I was blown away when someone pointed this out.