Have you had an adrenal stress index test? The A.S.I. saliva test can be bought from compounding pharmacies and sent to Diagnostechs in Washington. Iodine is helping your thyroid produce its hormone, but cortisol is needed to place the hormone into the cells. If your cortisol levels are low your hair will fall out even if you take care of iodine deficiency.
I am prescribed Cortef (bioidentical cortisol) because my morning levels and noon levels are too low. The thyroid needs cortisol. If you can get the saliva test it would reveal what time of day you need to take Cortef or whole licorice root (with glycerizzic acid content). One or the other, not both.