NewMexiKen has been provided with some background on the 22nd Amendment, including a paper by Bruce G. Peabody and Scott E. Gant, “The Twice and Future President: Constitutional Interstices and the Twenty-Second Amendment.” (Minnesota Law Review (1999), 83 Minn. L. Rev. 565).
The paper concludes there is no constitutional bar to a twice-elected president becoming president other than by election as president.
Particularly interesting to NewMexiKen however, was this material regarding Eisenhower.
As the election of 1960 neared, however, attention again turned to the Twenty-Second Amendment. In a press conference on January 13, Eisenhower invited reporters to look into the question of whether he would be eligible to run as a vice presidential candidate under the terms of the Twenty-Second Amendment. As Eisenhower put it, “the only thing I know about the Presidency the next time is this: I can’t run. [Laughter] But someone has raised the question that were I invited, could I constitutionally run for Vice President, and you might find out about that one. I don’t know. [Laughter]”
…
At a press conference two weeks after Eisenhower first raised the possibility of his serving as Vice President, he was asked whether he had received an “official opinion” on the question. Eisenhower was somewhat circumspect but he did say
that the afternoon of that [first] press conference, there was a note on my desk saying a report from the Justice Department–I don’t know whether the Attorney General himself signed this, but the report was, it was absolutely legal for me to do so. That stopped it right there, as far as I’m concerned.…
While Eisenhower ultimately backed away from the idea that he might run as Vice President, there is some evidence that, despite the constraints of the Twenty-Second Amendment, he did not completely relinquish his presidential ambitions at the end of his second term. Only four months after Kennedy’s inauguration in May 1961, Eisenhower indicated that he would have considered running for a third term if he had not been constitutionally barred from doing so and he had been able to foresee Nixon’s defeat in the 1960 election. Eisenhower’s son, John, also indicated that he and White House officials believed that had Eisenhower not been barred from running for reelection, he probably would have done so in 1960. And some political commentators have speculated that if Eisenhower had run, he would have been renominated and reelected.