In the lead up to the 1860 election, as the Democratic Party was split into factons Mississippi State senator Jefferson Davis suggested a compromise. From Generals in Blue and Grey - Volume Two:
"He proposd that Beckenridge accept the nomination [to the Southern-Democrates], hoping [Stehpen] Douglas would realize that, with a split in the party, his chance for election was near impossible. With this tactic, all three candidates would step down and allow a compromise candidate to be selected, one that the entire party could support. The plan was reasonable; both Bell and Breckenridge agreed, but Douglas did not. Douglas's refusal to participate in Davis' compromise left three candidates in the field to oppose the Republicans."
A possible scenario then would be for Davis' compromise to happen and for Douglas to be chosen as the only democratice candidate of the 1860 elections with Breckenridge as his second. Douglas only lived until June 3 1861 which would allow him to be President if elected but if, as I propose, Breckenridge takes up the office of Vice-President again he will take over when Douglas dies and so he would become President of the United States.
That however does not get him elected to the position.