One idea I have for a post 1968 election loss Nixon is that he runs successfully for Mayor of New York. Its kind of out there, but it would be a fun timeline.
One thing to remember is that the USA gets a Humphrey Administration, at least for four years, and the Great Society continues or is advanced. Humphrey also makes a number of Supreme Court appointments. He will at a minimum select the replacements for Harlan and Black (Rehnquist and Powell IOTL), likely for Warren and Fortas depending on what happens with the Fortas confirmation/ scandal ITTL (Burger and Blackmun IOTL), also likely for Douglas (Stevens IOTL) either due to getting a second term or Douglas moving up his departure. You also have a completely different set of cabinet appointments. Some of the Nixon cabinet members would run for elective office. As the administration party, the Democrats due worse in the 1970 elections and a lot worse in 1974.
At some point a Republican will be elected President, either due to Humphrey losing his bid for a second term, or more likely in 1976 due to a feeling that 16 years of Democratic administrations are enough. But its not clear if the Republicans will move to the right and go with Reagan, or to the left and go with Rockefeller or someone like him. The Great Society will have been vindicated in 1968. The Carter Administration is completely butterflied away. Probably also the GHW Bush administration, though ITTL Bush could well win his Senate race in 1970.
IOTL, Nixon was the most successful of the post World War 2 presidential runner-ups, running for the office later and getting elected. Looking at what the other post World War 2 losing candidates did afterwards, there seems to be two groups:
1. Senators: Goldwater, Humphrey, McGovern, Mondale, Dole, Gore, Kerry, McCain, Romney. With the exceptions of Mondale, Dole and Gore, this group either continued to serve in the Senate or, in the case of Goldwater, Humphrey, and Romney, ran for the Senate later and got elected. Also Kerry was Secretary of State and Mondale Ambassador to Japan. Dole and Gore left politics.
2. Everyone else. Dewey was re-elected Governor of New York. Stevenson ran again in 1956 and 1960 and was UN Ambassador.
Dewey re-elected Governor of New York in 1950, then retired. Dukakis left politics. The three presidents who lost re-election did ex-president stuff.
Its not clear which group Nixon would fall into, though if he goes back to the Senate the 1970 New York Senate election would be the best opportunity. And he might be re-elected in 1876 since its more likely a Republican year ITTL.