TSH w/reflex to FT4 means that they will only test for Free T4 (FT4) if the TSH is outside the normal range. And by any means, a TSh of over 20 would be outside of the range. So they most likely did the FT4 test. So you should be able to obtain that number and the reference range if you ask for it.
ALWAYS ask for copies of your test results. ANd/or demand that they give them to you verbally over the phone at the very least and write them down.
Furthermore I would recommend that you always demand that your Dr test for btoth Free T4 AND Free T3. many Dr's are reluctant to test for FT3. And will only ask for the reflex for FT4. But I would recommend that you demand they test each and every time regardless of what the TSH value is for both the FT4 and FT3!
Understand your body at the cellular level ONLY uses the Free T3 hormone. So if that is ultimately what your body uses to make you feel well. How insane is it to not test for the very hormone that is most important?
Yet this is what the medical industry teaches Dr's to do. And then wonder why so many people feel crappy and can't explain it. When they only get part of the story!
Yes, it looks right. When TSH is high, you are undermedicated, when low overmedicated. So, it looks like you needed the increase. However, FT3 and FT4 are much more important for managing thyroid meds than TSH is.
What was your FT4 with reference range?
They didn't test FT3?
Are you having hypo symptoms?