Here's a quick little scenario
Nixon narrowly wins in 1960. Come 1964, he faces the rise of an angry conservative branch within his own party. It isn't exactly going to wrest the nomination away from the President, though there may be some weak attempts to get Goldwater the nomination instead, which puts Goldwater and Nixon on less than friendly terms, though I somehow doubt Goldwater would be personally involved in this anti Nixon conservative group. In any case, the upsurge is at least somewhat vocal, leading to Reagan giving the speech he did IOTL as part of this conservative upsurge.
He then goes on to wins the Governorship as per OTL. Sorry if that seems unlikely and unoriginal, I just can't think of any other way to get Reagan politically involved in the given time frame. Come 1968, the conservatives having been denied the opportunity to get Goldwater the nomination in 1964 because of Nixon's incumbency, conclude that Senator Goldwater is too old to be their standard bearer in 1968, and they rally around Governor Reagan instead. The conservatives press hard enough for Reagan to actually get the nomination instead whoever Nixon would actually prefer as his successor. Meanwhile Senator Kennedy, the loser of 1960, having remained in the senate since his defeat eight years prior somehow manages to pull a Nixon, and gets the Democratic nomination a second time.
Reagan the standard bearer of the conservative movement faces off against the 51 year old Senator John Fitzgerald Kennedy.
sorry if the situation sounds a little twee, specifically the conservative upsurge at the 64' GOP convention, I just needed a way to get Reagan into a position where he could be governor and the heir presumptive to the Goldwater wing of the party. There probably a whole lot of problems with it, but you'd know them better than I would.