The Gemara praises a righteous woman named Kimchis because the "walls of her home never saw her hair". The Gemara ascribes to this merit that seven of her sons served as Kohein Gadol. Based on this, many Gedolim have advocated extra caution regarding uncovering ones hair, even in private, as a Segula for the children's success.
It is clear in the Gemara, and Reb Moshe also writes so in his Teshuvos, that this is merely a commendable Chumra and not a Halachic obligation. In this context, while it remains praiseworthy to cover her hair even in complete privacy, if it is uncomfortable to do so one is permitted to be lenient.