Just for info, Endocrinologists are not necessarily good thyroid doctors. Many of them specialize in diabetes not thyroid. Also, many of them have the "Immaculate TSH Belief" and only pay attention to that. Which is wrong. If they test beyond TSH it is usually only for Free T4 and if found within its reference range, they will tell you that you are okay. That is also wrong. I could explain in more detail if needed, but suffice to say that you should look for a doctor that meets the description I gave above. Typically those are hard to find. I am sending you a PM with some info that might help. To access, just click on your name and then from your personal page, click on messages.
Certainly you need to always test for Free T4 and Free T3 every time you go in for tests. Those are the biologically active thyroid hormones. Scientific studies have shown that hypo symptoms correlate best with Free T3 levels.
If not already diagnosed as having Hashimoto's Thyroiditis, then you should test for the antibodies of Hashi's. First test for Thyroid Peroxidase antibodies, abbreviated as TPO ab. If that is within range then they should also test for Thyroglobulin antibodies, known as TG ab.
When already taking thyroid meds like you are, testing for TSH is a waste of time and money. It adds nothing to the evaluation. In fact, hypo patients taking thyroid med adequate to relieve hypo symptoms often find that their TSH is very low in the range, or suppressed below range.
Looking at your test results, your Free T4 was at rock bottom of the range, and your Total T3 was actually below range. That is indicative of a med dosage that was inadequate. In your recent tests your Free T4 was adequate, but your Free T3 was not even tested. I expect that it is low due to inadequate conversion of the T4 to T3, resulting in hypo symptoms.
Most importantly, a good thyroid doctor will treat a hypothyroid patient clinically by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being constrained by resultant TSH levels. Symptom relief should be all important, not just test results. So you need to talk to your doctor about clinical treatment, as described, rather than dosing you based on TSH levels, which doesn't work.
Also note that hypothyroid patients are frequently too low in the ranges for Vitamin D, B12 and ferritin. D should be about 55, B12 should be in the upper part of the range, and ferritin should be about 70. If not tested for those you should do so and then supplement as needed to optimize.
So when you see your doctor I suggest that you request to be tested for Free T4, Free T3, Reverse T3, Vitamin D, B12 and ferritin. Also inquire about clinical treatment instead of treatment based on TSH level, and also ask if the doctor is willing to prescribe T3 meds like Armour Thyroid or Cytomel, if required to adequately raise your Free T3 level.