The Gemara in Ketubot 72a states that the source for a married woman’s hair covering is found in Parshat Naso in the section dealing with the Isha Sotah (wife who is suspected of being unfaithful) which would mean that it is a Biblical prohibition for a married woman to leave her home with her head uncovered.

It is written in Bamidbar, Naso 5:18: “The Kohen shall station the woman before God and expose the woman’s hair (ufara et rosh haisha), and place, on her palms the meal offering of remembrance; it is a meal offering of jealousy and there will be in the Kohen’s hand, the bitter, lethal waters.”

Rashi states that he disentangles the plaits of her hair to humiliate her. From this we derive that, for daughters of Yisrael, a bare head is debasing.

Rambam does not feel that a married woman’s requirement to cover her head is Biblical as it is only hinted to in the Torah and therefore he would qualify it as Rabbinic.

Tosafot explains that a married woman is not required to cover her head at home or in her own courtyard.

Rabbi Zeira explains in Ketubot 72b that if women were required to cover their heads even within their own courtyards “then you will have not left our father Avraham a daughter who will remain living with her husband!” In other words, if you are so strict then there will be no more reputable women left.

Some women may choose to be more machmir (strict) and cover their heads all of the time. The key here is that the women must have the opportunity to learn the leniencies and stringencies and make an educated decision of how they are going to observe the Halacha in a way that they are comfortable with.