He can't very well run for the Senate in 1962, though, given that Thomas Kuchel was up for re-election. True, Kuchel was detested by the right wing of the California GOP, but Nixon, at least as of 1962, wasn't a part of that (indeed, they ran a candidate, Joe Shell, against *him* in the primaries) and he is not likely to challenge an incumbent Republican.
1962 is one of the tricker scenarios, I was thinking more along RogueBeaver's line, but of course he could run for Senator in '62. The California GOP has already turfed a Senator and a Governor in 1958's Big Switch, they'll do it again if they (Nixon) really want to. I agree Nixon won't run in the primaries against him, but equally I suggest the GOP is more than powerful enough at the time to tell Kuchel to go run for Governor or something, because if he doesn't step down they'll cut him to pieces and he has a pair of recent fresh examples to remind him the Cali GOP & Nixon are ruthless.
Interestingly he may then lose the 1962 Senate race to some Democratic candidate, which would be a fun alternate take on OTL with the same basic result at least at first.
Senate: Nixon not becoming VP is the easiest route, but then it takes longer to become POTUS. He has to take control of the CAGOP, which might mean an even nastier bloodbath than OTL '58. Seniority-wise he could probably throttle Knowland after the '58 massacre like Halleck did to Martin and become leader then, but I don't see Nixon spending 20+ years in the minority.
Of course Nixon staying as Senator drastically changes the 1960 Presidential race so perhaps alt-Eisenhower VP takes it instead of Kennedy, thus forcing Nixon to wait at least until 1968 or longer if he still runs for President. By which time he's probably Minority Leader of the Senate and certainly things might have shaken out differently enough to come out with a GOP majority earlier than OTL. Or indeed he might just jump in anyway, but he'd lack the standing being VP gave him.
Or say Eisenhower alt-VP is a nobody then Rockefeller might get in earlier in 1960 and actually take the nomination. He'd easily beat JFK I believe, win in 1964, and probably get his candidate nominated in 1968 if the Goldwater wing can't get it together (if they can, well, 1964 scenario with equally matched side is a real nasty GOP fight) so Nixon is stuck until 1972 if the GOP nominee loses and 1976 if the GOP nominee wins.
If he never makes it to President I'll bet that Senator Nixon gets SecState at some point, but it is a very good question as to when and which President, because it is at best a poor fall back position for him if he doesn't get to be President. (Won't be Rockefeller of course, Kissinger gets to be SecState much earlier ITTL.)