The challenge here, with a divergence sometime after the election of Dwight Eisenhower as President of the United States, is to keep the Republican Party under the control of the Eisenhower/Dewey wing of the party over the long term, and to eventually make the GOP's ideology indistinguishable from that of President Eisenhower. In other words, the challenge is to make the GOP "the party of Eisenhower" rather than "the party of Reagan" ideologically speaking.
Given all of the organizational advantages the conservatives were starting to have by the late fifties/early sixties at the latest, this is a tough order, so good luck.