Thomas Jefferson eventually became President though...
Several Presidents fit the prompt by losing when they weren't incumbent. Jefferson, Jackson, Harrison, Nixon, am I forgetting any?
If we weren't excluding 3rd party candidates there's Van Buren, Fillmore, and Teddy Roosevelt. Teddy came in second place, so maybe at least he should count.
Teddy Roosevelt was actually the official Republican nominee in South Dakota, but by that logic Storm Thurmond and George Wallace would also qualify as major party candidates since they were both the official Dem nominees in Alabama.