"If S.P. Chase weren't in quite so big a hurry to be President, he would stand a much better chance."--Horace Greeley, 1858
In 1868, Chase might have sought the Republican nomination, but it was clear that the Republicans wanted Grant. OTOH, it seemed that he at least had a chance for the Democratic nomination: His conduct of Andrew Johnson's impeachment trial in the Senate had endeared Chase to many Democrats, and he was also beginning to become disillusioned with military Reconstruction. Chase summarized his own position on Reconstruction in 1868: "Congress was right in not limiting, by its reconstruction acts, the right of
suffrage to whites; but wrong in the exclusion from suffrage of certain classes of citizens and all unable to take its prescribed retrospective oath, and wrong also in the establishment of despotic military governments for the States and in authorizing military commissions for the trial of civilians in time of peace. There should have been as little military government as possible; no military commissions; no classes excluded from suffrage; and no oath except one of faithful obedience and support to the Constitution and laws, and of sincere attachment to the constitutional Government of the United States."
https://en.wikipedia.org/wiki/Salmon_P._Chase In short, "universal suffrage and universal amnesty"--but the problem with that is without the military, how do you enforce universal suffrage in the South? Anyway, this position was not really acceptable to either party in 1868. The most obvious problem in getting the Democratic nomination was stated by Chase himself: "it has seemed to me well nigh impossible to get over the difficulty induced by the almost universal commitment of the party to hostility to the colored people." (Quoted in John Niven, *Salmon P. Chase: A Biography* [New York and Oxford: Oxford UP 1995], p. 428.
As for his backing Vallandingham in the early 1870's, remember that Vallandingham was one of the originators of the "New Departure" movement in the Democratic Party
https://en.wikipedia.org/wiki/New_Departure_(United_States) which basically said that the Democrats should stop fighting the Civil War and the 13th, 14th and 15th Amendments, and get on to more profitable issues, especially economic ones. And on economic issues, it was easy for Chase to make common cause with the Democrats--remember that he was originally elected to the Senate by an alliance of Democrats and Free Soilers, and that one thing that made this alliance palatable to the Democrats was "his long-standing campaign to portray himself as an adherent to Jeffersonian states’ rights and to what had become a Democratic canon: tariff for revenue only, hard money, and opposition to centralized banking."
https://books.google.com/books?id=PGnmCwAAQBAJ&pg=PA120