Apart from having a good T3 level off the top of my head the common vitamins and minerals that cause hair loss are the ones that Thyroid UK tell those who have thyroid issues to get tested for.
1. Ferritin (iron storage protein) - your result needs to be around 80 or at least half way in range to be optimal and to actually grow. This is more than what most doctors who don't specialise in trichology recommend.
2. Vitamin B12 - your serum level needs to be at the top of the lab range
3. Vitamin D - your level should be around 100nmol/L. If your level is under 50nmol/L you may experience hair loss.
4. Folate - your level needs to be halfway in the lab range.
The one they don't mention that also is important is haemoglobin (iron rich protein that transports oxygen in red blood cells) - your level needs to be at or preferably over 12g/dL (120g/L)
While you need the micro-nutrients of zinc, magnesium etc if you are low in the main ones I've mentioned you have no chance of sorting out your hair loss.
I suggest you get a copy of your latest test results and check all these levels.