“I thought covering hair is a Halacha”
According to the overwhelming majority of sources, it is. It is a machloket whether it is d’oraita or d’rabbanan and among rishonim the d’rabbanans have a majority. There are a very small number of Acharonim who say that it is minhag.
What kinds of haircoverings are permitted (i.e. are wigs ok?), and how much hair must be covered, is a real machloket. My wife decides to be machmir in both areas; she wears hats and scarves that cover everything. I had the sense to keep out of her decision-making process. 😉