Conservative Hypothesis: Lincoln retires and Grant, a (former Democrat) popular war hero, is nominated with one between Schuyler Colfax, Benjamin Wade, Henry Wilson and Reuben Fenton as vice-president. He defeats Seymour and runs the country as OTL. Lincoln is before or later nominated to Supreme Court.
Changing Hypothesis: Lincoln leads to reconciliation with Southern states, promoting integration but without harsh military occupation, granting more funds for Reconstruction and not barring some of them from vote (in 1868 Texas, Virginia and Mississippi were still under militar administration). He fears Radical Republicans (as John Frémont) and Southern Democratic extremists, so he decides to continue the experience of Union Party. People as Grant, Johnson, Salmond Chase and Francis Blair would be all members of the party: I can see Grant be nominated with Blair as running mate (two popular war hero, one from North, one from South). Without Johnson, Blair and probably Winfield Hancock, Moderate Democrats unites behind Thomas Hendricks and nominates him against George Pendleton (Seymour's declination is accepted), with Sanford Church as vice-president. Radical Republicans bolts and runs with a Frémont-Wade ticket. Grant wins in a landslide. Union Party will become a big-tent centrist party while Democrats will be able to rebuild themselves and win in 1876 (without military occupation of South, Tilden would win). Radical Republicans, initially strong thanks its Congressional members, will collapse due corruption scandals and will divided between Liberals (that then reunite with Unionists) and Radicals, that under Benjamin Butler will make an alliance with Greenbacks and Populist movement, becoming the main rural leftwing party.