Trump brought an END to the Democrat Party....didn't need a "new" third Party....
The End of the Party
Henry Clay and others had called themselves National Republicans -based on their vision of the United States as nation while others saw it as a confederation of states - taking strong national measures like building inter-state roads. When a number of southern Democrats like John C. Calhoun, threw their lot in with the National Republicans, they were united only by their opposition to the growing "kinglike" strength of the president. Thus they came to be called Whigs, implying that the Jacksonians were Tories, in favor of "King" Andrew.
The Whig party ran, for some years, mostly in strong second place to the Democrats. They elected William Henry Harrison, in the famous "Tippecanoe and Tyler Too" campaign of nonsense, copied from the Jackson Democrats, but Harrison (the hero of Tippecanoe) died just days into his presidency, and was succeded by Tyler, one of the anti-Jackson democrats, who showed himself to be basically a firm Democrat, and was "read out of the Whig party". They also elected Zachary Taylor (another war hero and no politician) who was died fairly early in the term, making Millard Filmore president.
In the 1850s when the nation became increasingly divided over slavery, a new Republican party formed, primarily to keep slavery quarantined off in the South, while Southern sentiment was for their right to move, with their way of life, into any new territory. Their methods of agriculture and their best cash crops tended to deplete the soil, so that Southerners were among the most aggressive Western expansionists.
The Republican Party, while it also attracted many anti-slavery Democrats, drew off so many Whigs that they effectively killed the Whig party. The Whigs were also badly hurt by the short-lived Native American or Know-Nothing party, which was primarily anti-immigrant and anti-Catholic. This party was strong in urban areas, which had also been a Whig stronghold. The last year the Whigs had a presidential candidate was in 1856.