Does this reaction feed into the rise of the New Right's Evangelical wing later in the 1960s and 70s? Or was this spurred by the already-present roots of such political evangelicalism?

Bingo! And there is a book coming out this year that will just blow your mind if you're interested in this. Once it's released, I highly recommend you check out Ken Kruse's new book, One Nation Under God, which is basically a book-length answer to your question. In it, Kruse argues that it's a mistake to see the new political religiosity of the 1950s as a product of the Cold War. Rather, he shows how we can trace the origins of it back to the 1930s, and find its roots in conservative opposition to the New Deal. In Kruse's words:

This revolutionary transformation in America’s religious identity had its roots not in the foreign policy panic of the 1950s but rather in the domestic politics of the 1930s and early 1940s.

Decades before the inauguration of President Eisenhower, conservative religious leaders worked closely with corporate titans to plant the seeds for a powerful new union of patriotism and piety. As their private correspondence and public claims make clear, this new religious nationalism sought to challenge the state power that its architects feared most – not the Soviet regime in Moscow, but the New Deal administration in Washington. With ample funding from major corporations – including household names like General Motors, Kraft Foods, J.C. Penney, and Hilton Hotels – conservative clergymen and their political allies advanced a new ideology that was best characterized as “Christian libertarianism.”

In an impressive array of newspapers, magazines, radio programs, television specials, sermons and massive public events, these new evangelists for free enterprise argued explicitly that the New Deal was nothing but a program of “pagan statism,” one that violated all of the Ten Commandments. The Roosevelt administration, they argued, had essentially made a “false idol” of the federal government, leading Americans to worship it rather than the Almighty; it caused Americans to covet what the wealthy possessed and sought to steal from them; and, ultimately, it bore false witness in making wild claims about what it could never truly accomplish. Above all, they insisted that the welfare state was not a means to implement Christ’s teachings about caring for the poor and the needy, but rather a perversion of His doctrine. In a forceful rejection of the public service themes of the Social Gospel, they argued that the central tenet of Christianity remained the salvation of the individual. And if any political and economic system fit with the religious teachings of Christ, it would have to be a system rooted in a similarly individualistic ethos. Nothing better exemplified such values, they insisted, than the capitalist system of free enterprise. In the end, the politicization of religious conservatives was essentially an unintended consequence of the revolt against the New Deal.

Source - note that this is from an abstract of Kruse's book rather than the book itself, as it's not officially in print yet. Having been to a couple talks he gave about the book, and read some of the excerpts/chapters that have been floating around, though, i can tell you that he makes a very convincing argument. I expect it to be one of the most talked-about books of the year.

There are, I think, two contributing factors.

Building on /u/AnacreonInHeaven's answer, the roots of fundamentalist/evangelical thought at the beginning of the Cold War can be traced back to opposition to the New Deal. The ideological framework supporting this opposition was theological, coming from dispensational premillennialism. One of the tenents of premillennial thought was the idea that before Christ returned, the Antichrist would form a totalitarian regime, taking power from world governments, as part of the End Times. Evangelicals were therefore pretty skeptical of anything that sought to centralise state power.

Matthew Avery Sutton's new work, American Apocalypse, details precisely that argument, so you might find that interesting.

The second factor was Truman (rather than Eisenhower). Truman set up the ideological framework that Eisenhower's administration brought to fruition. Truman, himself an evangelical, faced a difficult task mobilising support for opposing the USSR. There was a clamour for a return to the traditional policy of isolationsism (see, for example, the abrupt end of Lend-Lease), but Truman's foreign policy objectives required unprecedented international involvement, for America to take a leadership role in world affairs.

There was also a growing mistrust of communism, which had been abrogated during WWII, but resumed almost immediately following its conclusion. This included things like HUAC and McCarthy's Senate hearings. For Truman, this was an excellent opportunity - take that existing anti-communist feeling, hype up the dangers of communism, and use that as a reason for the US to keep involved on the international stage. The only problem was that anti-communism was a rather negative form for an ideology to take.

In order to compete, Truman helped shape the new ideology in religious terms. The struggle against communism was recast as a Manichean conflict between good and evil. There was a strong element of crusading evangelicalism in propaganda of the time, and discourse took on a decidedly religious flavour.

This achieved several important objectives with a single move. It created a positive doctrine for the West that emphasised the values for which it purportedly stood. It also provided an ideology that could be used to compete with communism, Christianity serving as a unifying factor that could appeal across the Western world. Crucially, it also allowed the West to exercise claims of moral superiority by attacking the perceived Achilles’ heel of communism, its atheism.

Allied propaganda had successfully cast the Second World War as an existential battle between the forces of good and evil, paying particular attention to the more pagan elements of Nazi ideology. By similarly attacking the avowedly atheistic Soviet Union it aimed to harness much of the same energy and portray the struggle as a crusade against the godless, and thus implicitly evil, communist regime and its ‘fanatical pseudo-religion’.

Those are the two things that came together to create the religious turn that you describe. There's an excellent edited volume (by Dianne Kirby) called Religion and the Cold War which would give you a good overview and some specific essays that would further develop those themes. I'd also suggest Kirby's article 'Harry S. Truman's International Religious Anti-Communist Front, the Archbishop of Canterbury and the 1948 Inaugural Assembly of the World Council of Churches’ in Contemporary British History xv:iv which discusses the necessity of a common ideology between the western nations (particularly the US and UK) and argues that religious terminology and the role of prominent churchmen was a decisive factor.