Tracking your cycles. Either with natural cycles or flo x
You can use a diaphragm with spermicidal lube
How important is it that you don’t get pregnant? And why do you not want hormones? We’re using natural cycles but if we get pregnant accidentally I’m ok with that. That’s the only context I would use it in. Copper coil is effective but can make periods heavier, something to consider if your periods are heavy already. Mirena coil has hormones but because it’s right next to your ovary it’s really low dose of progesterone only, worked really well for me in the past. Otherwise you have barrier methods, condoms/diaphragm ect.
Copper coil. Doesn't use hormones. I asked my GP about non-hormonal contraception, and that's what they recommended https://www.nhs.uk/contraception/methods-of-contraception/iud-coil/what-is-it/