ATP cofactors is the reccommended supplement for use with iodine
Healthy fats are a very good idea while you're going through the detox phase, the thyroid uses them very much while making it's hormones and detoxifying the blood.
I suggest switching from the Calm brand Magnesium to a supplement containing Magnesium Taurate or l-Threonate when the bottle is empty. The taurine/threonate amino acids strengthen the body!
While you are detoxing, the body needs to redbuild many tissues. Healthy fats are the some of the best tools for the job!
Oil-pulling is one of the ways to get these healthy fats in, together with an increase of fruits and veg in your daily diet. Cold-pressed extra virgin olive oil, thistle oil, and coconut-oil are some of my very favorites. :)
The Iodine cofactors really help with making stomach acid. (excellent article on the importance of the stomach's ability to make hydrochloric acid, with explanation of the nutrients needed for it: http://www.tennantinstitute.us/conditions/stomach-acid.html )
Have you got zinc and B1 in your supplemental regimen? They may be beneficial.
In the long run you may save a great deal of money by investigating the liquid forms of iodine, which are primarily Lugols' and super-saturated-potassium-iodide (AKA SSKI ). Trapper offers bottles of SSKI often, and there are several others here who provide good products. Iodorol is a tablet designed off of the formulation of Lugol's iodine, which is 2 parts potassium Iodide to 1 part unbound iodine.