Eisenhower certainly was a Conservative, but hardly a member of the far-right. He had a real sense of playing to the middle, and expressed disdain for extremists on both sides. At one point, he said about his nomination of Wareen to the Supreme Court, that one of his brothers had written him claiming that Warren was a comunist, and another brother had written complaining that the California governor was a reactionary; Ike joked that if he was being attacked from both sides, he must be doing the right thing (of course, Warren's transformation from a centerist as governor to a liberal on the bench caught Eisenhower by surprise).
As President, Eisenhower saw himself firmly opposed to the Midwest conservative elements within the party (i.e. the Taftites) due to their isolationist foreign policy as well as their economic and social goals (he was, generally, a bit more sympathetic in these matters, but still felt that they were unlikely to coem to fruitition). To Eisenhower's mind, he appears to have felt that one can not turn back the clock, the New Deal had happened and could not be undone. Furtermore, as the cold War deepened, Eisenhower was not completely opposed to a stronger federal government, as well as it was well managed.
The problem is that Eisenhowerism was never met with a great deal of excitement by the party base, or those Republicans in Congress (Republican governors, who generally seemed to be more liberal than Republican congressmen during this era, were a bit more inline with Eisenhower's ideas).
If you are going to get a more moderate Republican Party, I think you are going to have to keep the Eastcoast wing of the party firmly in control; which means no influx of southern conservatives, and the John Birch-ilk. Having Nixon win in 1960 is a good start; Nixon was a moderate himself (in fact, there are those who claim Nixon was the last President of the Liberal Consensus). If Nixon is able to win two terms, that gives us 16 years of moderate Republican rule, and might have some last impact upon the party's ideology.