In 1968, the Dixiecrat segregationist George Wallace ran for president under the American Independent Party ticket. His main objective was not to become president as a few misunderstand. He knew such a campaign would be doomed to fail. Instead, he sought to achieve two goals:
1. Gain the electoral college votes of the southern states
2. Divide the conservative vote in the northern states, thus letting Hubert Humphrey win key states by pluralities.
If everything went according to plan, no candidate would gain the needed 270 votes to win the electoral college. The election would go the House, where he could broker a deal with one of the other candidates to give them the needed votes from the Southern state delegations in exchange for an end to federal efforts toward desegregation.
Let's play with this scenario: Say, Wallace managed to run a more effective campaign, and he wins the states of Louisiana, Alabama, Mississippi, Arkansas and Georgia, as he did IOTL. But he also wins the states of Florida, South Carolina, North Carolina and Tennessee. In addition, he manages to gnab just enough voters from Nixon in Missouri to allow Humphrey to gain the state by a plurality. The electoral college would be divided between Nixon with 244 votes, Humphrey with 203 votes and Wallace with 91 votes. With no one winning the minimum 270 votes, the election goes to the House.
What would happen next? Obviously no one's going to elect Wallace for president, but would Nixon or Humphrey be willing to cater to Wallace's demands in exchange for the presidency?