All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
It is prohibited for the menstruating woman to touch the Mus-haf with her hair or any other part of her body directly without a barrier. The Hanbali book of Fiqh Sharh Daleel At-Taalib reads, “It is impermissible for the person who is in a state of ritual impurity to touch the Mus-haf with his skin without a barrier.”
Ar-Ruhaybaani wrote, “It is also impermissible to touch the Mus-haf or part of it ... with his hand or another part like the chest, as touching implies having any part of the body in direct contact with it. Hence, it is prohibited for him to touch the Mus-haf even with his fingernails, hair, or a tooth that is still attached to his body, out of respect for the Mus-haf. This is a strong scholarly view.” [Mataalib Uli An-Nuha]
There is no harm in touching the Mus-haf with a barrier according to some scholars. The Hanbali scholar Ar-Ruhaybaani wrote, “It is not prohibited for the person in a state of ritual impurity to touch the Mus-haf with a barrier such as a bag or sleeve because he is actually touching the barrier in this case. It is also not prohibited for him to turn its pages using the barrier.” [Mataalib Uli An-Nuha]
Therefore, it is impermissible for the menstruating woman to let her hair touch the Mus-haf directly.
Allah knows best.