hymenal tissue, insertion, removal

So I am so excited to own a menstrual cup and have used it a few times successfully, but I just tried using it today after not having used it for the past two months and I felt the need to whinge.
Without fail every time I use it, hymenal tissue/skin around the entrance to my vagina gets pinched and it fucking hurts. This happens on insertion and removal. This makes me annoyed because I now dread inserting it because when I'm feeling sick, am in pain and am irritable the last this I feel like doing is pinching my delicate hymenal tissue. I also resent the feeling of the cup opening up inside me. Not welcome in a nonsexual scenario.
P.S. I have been sexually active for ten years, so I do not have an intact hymen. The skin there is not an issue in any other circumstance.