Somebody is probably writing a thesis about that right now. It's complicated.
In 1854 former Whigs ran as the "Opposition Party" and they actually won a majority in the House of Representatives. This group became the Republicans. In 1856, they nominated Fremont and came in second in the presidential elections.
Other Whigs joined the "American Party," also called the "Know-Nothings." The Whigs didn't quite die out yet -- Millard Fillmore the "Know Nothing" candidate for President in 1856 ran on the Whig label in many states, and the Constitutional Union Party of 1860 was considered by many to be "the Whigs" -- but they were pretty much a "spent force" by 1854.
In any case, so far no party has gone from being an established third party to one of the two major parties. There has to be a collapse or disappearance of one party for a new one to replace it.
I can think of two "major" parties that need to be "collapsed" right now.
It was essentially a merger of the Whigs who didn't join the Republican Party with the American Party (the "Know-Nothings").