So, by definition, a Democrat would have won in 1864. If said Democrat runs in 1868 and wins the Dems are on pretty solid ground for another running in 1872, and IMO, the Republicans would splinter in 1868 along Radical and Moderate lines, leading to a three way race which probably propels another Democrat to power in 1868. There's a few options after that, firstly the Republicans could set aside their differences to come together in 1872, while conversely they could go on separate courses but one faction could come to power. Second option is the Republicans unify, but a third party coalesces around issues both the Democrats and Republicans have a hard time addressing (like OTL's Populist Party) which might emerge sometime in the 1870s-1880s.
I honestly think its a seesaw between Dems and Reps till the 1880s when a new faction like the Populists comes into play, I voted socialists because some kind of organized labor movement emerging in the 1880s seems logical to me. Whether this would be an outgrowth of an unlikely alliance between rural and urban workers or a purely urban power is up to the reader.