Hayek clearly explains that this is not the case. That societies and their moral codes evolve together.
I agree that Hayek is very convincing on this, he does a good job of explaining why the "rationalism" that is the root of socialism is so flawed.
However, IMO, Objectivism does agree on many points with Hayek. However, I have to admit to not being very knowledgable on Objectivism, I am making surmises based on my recent reading of The Fountainhead, which I enjoyed.