I agree that you need the Free T3 and Free T4, along with the TSH test.
It's also good to get the antibody tests to determine whether or not you have Hashimoto's. Another thyroid ultra sound would also be warranted.
Just to clarify -- the TSH test is extremely "accurate" and will give a pretty exact level of TSH, at the time of the blood draw, so it's not the accuracy of the test that misses some cases of hypothyroidism. The problem is that TSH does not tell the whole story, because TSH levels can be "in range", while FT levels are too low for the individual. Additionally, too many labs, therefore, doctors, as well, don't use the new (10 yrs old) reference range recommended by AACE, so many people with TSH between the recommended upper level of 3.0 and the currently used upper level of 4.5 to 5.5, often get misdiagnosed.
The best indicators are Free T3 and Free T4. Notice it is Free and not Total, which is a different and less useful test. Free T3/T4 will show the levels of usable thyroid hormones in your body. TSH is only slightly useful; more of a diagnostic tool than anything, and as you probably suspect, it is not always accurate enough to catch all cases of hypothyroidism.
Other important tests:
thyroid antibodies-- 2 tests, TgAb and TPOAb
vitamin B12
vitamin D
ferritin and full iron panel
When you get your results, please post them with reference ranges (they vary from lab to lab).
Do you remember what your Synthroid dose was before when you were taking it?