I supplemented with iodine, in my case lugols iodine, as I had other issues as well due to iodine deficiency. Hv recommends kelp and other iodine rich foods, which is also fine with me, but lugols (and the companion supps) worked for me. I was extremely iodine deficient, among other deficiencies, including magnesium, potassium, vitamin B (all of them), and others.
We have to keep in mind though that iodine does not work for all causes of hypothyroidism, and that high doses of iodine can aggravate Hashimoto's thyroidosis.