AHC: Create another G. Cleveland!

How about Nixon in '56? Ike very nearly didn't run for reelection in OTL. Suppose he decides not to run. Despite Ike's private opposition, Nixon defeats Bill Knowland for the Republican nomination, picks Christian Herter as VP and defeats Stevenson in the General Election.

Four years later in 1960, Nixon gets stuck with the baggage of the '58 recession, Sputnik, and the Cuban Revolution. Kennedy beats Nixon, making Nixon the youngest ex-president (just 46) in history.

Assuming things unfold as in OTL, eight years later in 1968, Nixon returns and claims he's the only candidate capable of restoring order. In '72, he backs Connolly behind the scenes, but ultimately Reagan defeats both Connolly and Agnew then beats whoever the Democrats nominate.

So you get:

34. Dwight D. Eisenhower (R): 1953-1957
35. Richard M. Nixon (R): 1957-1961
36. John F. Kennedy (D): 1961-1963
37. Lyndon B. Johnson (D): 1963-1969
38. Richard M. Nixon (R): 1969-1973
39. Ronald Reagan (R): 1973-1981

And beyond that, the butterflies grow too big. But that may be a possibility.
 
Gerald Ford runs again for the White House in 1980, narrowly beats Reagan in the Republican primary with the help of the Republican establishment, and then goes on to defeat Carter with relative ease in the fall.

His second term is uneventful. The Republicans have the Senate, but they lose it in 1982 and Ford is unable to get through the same tax cuts and deregulatory bills Reagan achieved as OTL. Recovery is slow, Ford is a lame duck from the midterms forward, and Ted Kennedy wins the White House in 1984 on a promise to put America back to work.
 
You forgot about the 22nd Amendment, not to mention Ike's health.

1956: Truman was 72 and would not run again. While Truman is revered on a bipartisan basis today, his approval ratings in 1952 were lower than Nixon's in August 1974 and was viewed as a Democratic version of George W. Bush: bungling a war, a sluggish economy, and a midterm massacre at the 6-year mark. That didn't begin to change until the 1970s.

Bush 41 is too old, in too poor health and was never liked by his party's base even though he played a key role in founding the modern GOP in 1962.

Carter: the party would never renominate him and never liked him. In 1978 less than 60% of Democrats approved of Carter's performance: partly because he was the first New Democratic president domestically, partly because of his arrogance and obstinacy in dealing with Congress. IOTL he tried to garner some support for this but it was quickly shut down by the Democratic supers.

HHH was never a strong candidate: Big Labor's darling but like Lyndon Johnson, possessed of a political style that was obsolete in the 1960s, let alone the 1970s, too weak a personality to grab the prize.

The closest bet is Ford, but Ford was too centrist for the party in 1980 and if he somehow won, like Bush, he would be spending a lot of time fighting the base on domestic policy.

Working on your Ford idea, though it may not count under the exact terms of this AHC, I can get Ford back in without too much trouble: Ford accepts Reagan's VP offer in 1980 (IIRC, they tried to "do a deal" and it went nowhere; assume that Ford jumps at the deal instead), and Reagan is either killed or incapacitated in the '81 assassination attempt.

While not elected twice, you'd have Ford in office twice without ever being elected President.
 
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