Postmodernism’s rise is hard to pin down. Where existentialism has clear “leaders” that tell us what their philosophy is a response to, we don’t really have that with postmodernism.
However, I do see it as a building block, where existentialism identified meta-narratives in our interpretation of ‘meaning’, postmodernist see those meta-narratives everywhere. An easy example is in science, where many believe it to be objective observations, postmodernists see metanarrative within the interpretation of those observations.
So I would be willing to claim that postmodernism is a response to society’s reliance/belief in the objectivity of science and our reliance on technological innovation.