I don't consider magnesium to ever be a bad player. Sometimes the form of magnesium is an individual issue, such as I can't take magnesium taurate or magnesium aspartate, but magnesium citrate, malate, and bicarbonate are good for me, personally. It can be so hard to hold onto to magnesium that I think most people should take as much as their gut will allow -- you don't want diarrhea from it.
Some time when you test TSH again, also test RBC magnesium. It should be at least 6.0 mg/dL.
Sometimes hair loss is related to being hypothyroid or having low iron. Invariably when I start losing more hair than normal, my iron is low.
Sometimes skin issues are helped with zinc. You might consider using 25-30 mg of zinc picolinate or citrate. You may also need 1-2 mg copper to balance, taken at a different time from the zinc.
Generally, sun and salt water can be very beneficial in healing skin.