Is there any Shut sefer or other that permits a women not covering her hair NOT the Aruch Hashulchan saying it is not Ervah but allowing a women to go out in modern times with their Hair uncovered?
To summarize, Rabbi Yosef Messas (pronounced "Mashash") (born about 100 years ago in Morocco, served as rabbi in Northern Africa and then later in Haifa) allows it in a responsum, Mayim Chaim II OC 110:
הראת לדעת שהדבר תלוי במנהג, ואם כן בזמן הזה שכל נשי העולם בטלו מנהג הקודם וחזרו למנהג פשוט לגלות את ראשן ואין להם בזה שום חסרון צניעות
Thus it is stated that the matter depends entirely on practice, thus today when all the women of the world have negated the previous practice, it is simply allowable to uncover their heads, this reflects no lack of modesty.
Though some claim he was only referring to specific communities, where if he didn't allow it, they would stop keeping Torah, etc.
This is one of the many, many sources cited by Rabbi Michael Broyde in an article in Tradition 42:3 (2009), in which he argues as a limud zechut -- to judge favorably on the many noble women who sacrificed (and still do today) tremendously to keep Torah, Shabbos, kashrus, taharas hamishpacha, etc., but did not cover their hair.
Rabbi Broyde argues that the requirement for any hair covering for married women is culturally relative; he has a few acharonim such as the above, plus some readings of Rishonim on various Gemaras to support that view. He then goes further to suggest the Tur and Shulchan Aruch (whose wording on the subject follows the stream-of-consciousness style of the Gemara, not the clear reorganization that Rambam uses) could be read that way as well; on this point, many disagree.