It's a strange phrase indeed. It's from a foreign phrase. There's a period where a slogan "hats lead the society to become a superpower" was in effect.
*Edit I think I should have given historical background. The producer of the slogan wanted to modernize the society in order to make it more powerful, and thought that wearing hats was one way to modernize it. Of course, the slogan was absurd and doesn't make much sense. This absurdity makes people give the period a nickname to mock it, using a part of the slogan and the idea: "society which is led by hats" era - "hats-led society" era.
I'm not sure if "hats-led society era" or the variation is grammatically correct, or if there are other phrases that better convey the meaning.