I've recently lowered my dose of thyroxine. In fact, I've moved back to thyroxine only, no natural thyroid for the moment. I noticed I was getting hyper when I noticed that I was very low in iron. I took iron tablets and iodine and other good things I was already taking at I quite quickly got hyper. So I am on a lower dose and will get my TSH checked in a few weeks to make sure it's good. I have read before that iron is a critical nutrient for thyroid health. Additionally, I find it's critical for managing autoimmune thyroid.
I was just doing some reading and found this alternative treatment for hypothyroidism. It is controversial, and I am not recommending it. Talk to your doctor about all treatment and be aware that this is primarily an iodine supplement, and it is possible to overload your body.
Please note: it is possible to be allergic to iodine (very rare).
Disclaimer: for medical advice contact a medical professional.