I agree with most of what has been written above, however Eisenhower apparently considered starting a 3rd party:
"Ike threatened to become an independent if the Old Guard fought Earl Warren’s appointment as Chief Justice of the Supreme Court."
https://www.dwighteisenhower.net/partisanship-and-ideologyAlso, I believe I once heard that Eisenhower offered to run as a Democrat to their party leadership if it appeared likely that Douglas MacArthur would win the Republican nomination.
Most likely that Eisenhower would have tried to use his fame and status to start a centrist third party given the Republicans of today (and going back as far as 40 years.)