Since the Republican Party of the 1850s and 1860s was more-or-less the antislavery Northern wing of the Whig party, you just need to change the way the split plays out so the Northerners keep the party label rather both wings of the party leaving (IOTL, northern Whigs left first, mainly ending up with the Republicans, while southern Whigs largely reverted to the older "Oppositionist" label once it had become clear that the national Whig party had become irrelevant).
One scenario: Winfield Scott doesn't seek the Whig nomination in 1852. Southern Whigs back Fillmore as IOTL, and Northern Whigs back a free soil candidate (Fremont?). The convention deadlocks, and Southern Whigs eventually walk out and nominate Fillmore on a separate ticket.
Alternately, have the proto-Republicans in 1853-4 decide to organize towards a capture of the part of 1856 Whig convention rather than splitting off and starting their own party.