This may be marginally ASB but here goes:
Lincoln dies of typhoid fever 2 weeks after being nominated. As Vice-Presidential Nominee Hannibal Hamlin claims the Presidential nomination but as a regional poitical figure with no independent stature his calim is disputed. The National Republican leadership, under the prodding of William Seward, accepts Hamlin's claim but names Seward to the ticket as VP. This energizes midwestern and western devotees of John Fremont to start a splinter ticket, with Oliver Morton of Indiana as his VP, thus breaking the Republican Party into essentially A New England/New York branch and a Midwestern/West branch. In the confusion, John Bell drops his campaign and endorses Douglas. The Bell Douglas fusion and the Hamlin - Fremont split enables Douglas to carry the border states, NJ, PA, Ohio, IL CA and Oregon with 127 EV. Hamilin holds New England and NY with 78 EV, Breckendrige keeps the deep South with 61, and Fremont ends with Indiana, MI, IA, WI and MN for 30. This eliminates Fremont and sends the election to the House.
The House (1858 election), lacking a true majority, is hopelessly deadlocked. The Senate (1858), with a Democratic majority, sees that majority reunite to elect Douglas's VP, Herschel V. Johnson of GA, as Vice President. This is accepted by both wings of the Democratic party so the Deadlock in Presidential voting is alowed to continue and on March 4, 1861, Herschel V. Johnson is sworn in as Vice President and then as President of the United States. Under these circumstances, South Carolina does not secede in December 1860 nor do any other Southern States. However, Hamlin and Fremont Republicans alike view the entire scenario as electoral theft, worse than the Adams/Clay corrupt bargain. Needless to say the Johnson administration is a worse fiasco than Buchanan's and the two sides are spoiling for a return engagement in 1864, this time with no compromises.