I’m not a doctor and have no medical training whatsoever. I do, however, have Hashimoto’s, and here’s what I recommend based on my own experience: you should have your T4, T3, and reverse T3 checked. (Most doctors do not run T3 and reverse T3, even though T3 is the only metabolically active thyroid hormone.)
You want your T4 and T3 to be on the HIGH end of the normal range. For example, if normal is 40-180, you want yours to be around 150-180. I find I just don’t feel good if my T3 is in the low end of the normal range.
You want your reverse T3 to be super low, or negligible. Reverse T3 masquerades as T3 (but doesn’t actually do anything as far as thyroid support goes), so if your reverse T3 is high, the T3 reading will be way off.