Before 1960? Well, let's see: President Jefferson had to send Marines to deal with Muslims who thought their faith justified preying on shipping in the Mediterranean, a over a century before that, they were knocking at the gates of Vienna. (Think that's irrelevant? 9/11 was chosen as the anniversary of their last big defeat.)
Just before 1960, they were being imprisoned and executed by dictatorial Arab governments, mostly ones within the Soviet sphere of influence, or exiled by the Shah (in our sphere of influence). Qutb and the Muslim Brotherhood were ruthlessly suppressed by Nasser, and the Ba'athist goverments in Syria and Iraq stepped on theirs pretty heavily, too. Before the 1970's Lebanon was about 40% Christian, with the rest divided between Sunni, Shi'ite and Druze, so the crazies got no traction there.
Until the 1967 Six Day War dashed Arab hopes of 'pushing Israel into the sea' the Arab governments could credibly harness some of Islamist tendencies to their own 'anti-Zionist' project.
Of course the Chechen rebellion and Taliban didn't exist then either--though the Chechens had fought the Tsars, who had a less heavy hand than the Soviets.
You're also overlooking the Muslim lunatics on the Indian subcontinent, whose main activity are attacks on India, which always tilted Soviet during the Cold War, and is still friendly with Russia.
Islamism is its own evil. The left in the West have become the Islamists useful idiots through their embrace of 'multiculturalism', even as they were useful idiots for the Soviets because they blinded themselves to the horrors of Communism by the specious fancy that it was akin to social democracy (which also doesn't work, but at least doesn't slaughter millions to try to prop itself up through terror).
The fact that our opponents on the national political field are siding with both sets of enemies should not blind us to the fact that Islamism (or Islamic fascism, both are just names) is a distinct phenomenon.
The present war is not a continuation of the Cold War, but of a resumption of old wars from long ago.
Okay. That sounds reasonable and correct.