Do some schools of thought, such as the Hanafi school, permit a woman to expose part of her hair from under her headscarf?

None of the jurists have deemed it permissible for a woman to intentionally uncover even a small portion of her hair to a non-mahram man, for a woman's hair is considered awrah (private, to be covered) by the consensus of jurists. The Hanafis even held that it is not permissible for a man to look at what has been separated from a non-mahram woman, even if she is deceased.