With a POD of the 1950s, your goal is to have an incumbent President Barry Goldwater trump Lyndon Baines Johnson in a Presidential election. Preferably in the 1964 election, also bonus points if Goldwater manages to paint Johnson as a Anti-Civil rights war-monger, threatening to bring the world to nuclear war.
IMO the easiest way is to somehow get a Nixon-Goldwater ticket in 1960 to win, followed by Nixon's tragic assassination in 1963. Goldwater and many new Republican congressmen suddenly cruise into office.