The best approach is to initially test for TSH, Free T4, Free T3, Reverse T3, cortisol, Vitamin D, B12 and ferritin. Since you are already taking thyroid med, the tests needed every time you go in are Free T4 and Free T3. Testing for the others depends on initial test results and action taken in response. For example if your ferritin was less than 100 and you have been supplementing, then you will want to continue to test periodically until optimal. Doctors have been taught that TSH is basically all they need to know to diagnose and treat a hypothyroid patient. That is very wrong. Diagnosis and treatment should be based on symptoms that occur more frequently with hypothyroidism first, supported mainly by tests for Free T4 and Free T3.
You should also be aware that often hypothyroid patients taking T4 med find that their body does not convert T4 to T3 well, and Free T3 levels remain too low to relieve symptoms, unless the T4 dosage is high enough to raise FT4 to the high end of its range or even above range. A better approach is to add a source of T3 to the meds, to get Free T3 into the upper half of its range, or as needed to relieve hypo symptoms.
What is your daily dosage of Synthroid? Please tell us about all the symptoms you have. Please post your latest thyroid related test results and reference ranges shown on the lab report. If also tested for cortisol, Vitamin D, and ferritin please post those as well.