i go to a mixed gym with mostly non-Jews and dress modestly while there. However I find it very difficult to work out with my hair fully covered. In the community I live many women don’t fully cover their hair. Would it be permissible for me to wear a baseball cap with some hair exposed?
Aside from some loose hairs etc., all hair should be covered in such a public setting. The fact that many women in your area don’t properly cover their hair would not change the halacha for you.