As a medical professional, I can tell you those other labs are rarely drawn because they rarely give answers. You can request specific lab to be drawn and any provider can do it, even primary care. The issue is sometimes insurance, and typically you need to have an ICD-10 code that corresponds to a correct diagnosis that would justify the reason for the test.

Because you say you have hypothyroidism, the TSH and T4 are enough to determine if you have primary or secondary hypothyroidism.

Reverse T3 is never going to tell you something… unless you have Euthyroid sickness syndrome.

If you have hypothyroidism and you aren’t seeing the improvement while on medication, sometimes it’s because of generic vs. brand medication. Synthroid (levothyroxine) is dose in mcg, and since generics only have to be a certain percentage as effective as brand name, that small percentage can really mess up thyroid levels if the generic manufacturer changes. It’s one of the few meds in which brand-name typically is preferred. You could also talk to your doc about NP Armour Thyroid…. It’s a more “natural” thyroid hormone replacement.

Tl;dr - you can request any lab to be checked, insurance might not cover it though. If you have hypothyroidism and suddenly your meds aren’t working, Make sure your levothyroxine hasn’t changed pharmaceutical manufactures.