So a common rejoinder by modern Republicans to accusations of racism is that Lincoln was a Republican and he freed the slaves, while the KKK was founded by Democrats who defended slavery. This is technically true, but it ignores the fact that the parties basically swapped ideologies, especially in regards to race relations and civil rights. (https://en.wikipedia.org/wiki/History_of_the_United_States_Democratic_Party#The_South_becomes_Republican)
This disingenuous argument wouldn't be possible if they had instead formed two new parties with new names, and it would have avoided a lot of the confusion that has happened since. Why didn't they do so? Was this possibility considered?
The Fifth Party System is generally thought to span 1932-1964. The Sixth, current, system dates from then: the passing of the Civil Rights Act in 1964. Ofcourse it was a much more gradual process than this suggests. It wasn't really finished until Newt Gingrichs Republican revolution in the 90s.
US history is littered by failed third parties: the People's Party (1892-6), Theodore Roosevelt's Progressive Party (1912). First Past the Post is perhaps the main reason that third parties struggle in America. Without one of the dominant two collapsing, as happened with the Whigs in the 1850s from which the Republicans emerged, a third party is effectively locked out of power.
Republicans had been courting disaffected Dixiecrats for some time, particularly Barry Goldwater in 64, and Nixon continued to make inroads. Despite this, there were efforts to create a new party. The Dixiecrats formed a shortilved party for themselves in 1948. In 1967 the American Independent Party (AIP) was formed to promote Alabama's George Wallace's candidacy and succeeded in carrying 5 states in the 68 election. Nevertheless the Republicans were the obvious choice for conservative Democrats. The 1972 landslide for Nixon confirmed this and was followed shortly after by the AIP splitting.