The doctor said "normal" because your result was within the reference range. Unfortunately reference ranges are typically based on statistical analysis of test results from a sampling of people; therefore, a range may not represent what is best for a person. Have a look at this link from the Vitamin D Council. It suggests 50 should be the target.
https://www.vitamindcouncil.org/about-vitamin-d/testing-for-vitamin-d/?gclid=CIynofnx1MwCFQmqaQodqMEMvQ
You can find this and other useful info about hypothyroidism in the following link. I recommend reading at least the first two pages nd more if you ant to get into the discussion and scientific evidence supporting all the recommendations.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
That Vitamin D level is too low. Should be at least 50, so you should discuss with the doctor, supplementing with about 2000 IU daily.
What was the Vitamin D result? Just being in the low end of the so-called "normal" range is not adequate. Also, what was the calcium test result?
Symptoms are the most important indicator of thyroid status. So please tell us about any symptoms you have.
Having had a TT, your dosage of Armour apparently was not adequate as a full daily replacement amount. As a result your Free T4 is too low, and I suspect your Free T3 is also inadequate; however you were tested for Total T3, not Free T3. Free T3 is the biologically active thyroid hormone used by all the cells of your body. You need to make sure they always test for both Free T4 and Free T3 every time you go in for tests.
Clearly your Free T4 is too low in the range and I suspect that your body is converting extra T4 to T3 in an effort to maintain thyroid function as well as possible. You do need for your Free T4 to be increased, but it could be that instead of adding Levo to your med, an increase in Armour could suffice. To better assess the possibility, you do need to test for Free T3. So, I am not sure why the doctor would instead order tests for thyroid antibodies, T3 uptake, and Free T4 again. I suggest that you should question that and ask for Free T3 instead, or at least in addition to those if he doesn't want to change.
Since hypothyroid patients are so often deficient in Vitamin D, B12 and ferritin, I also highly recommend asking for those as well, and then supplementing as needed to optimize. D should be at least 50, B12 in the upper end of its range, and ferritin should be at least 70. If you can get these and the Free T3 done, we can help interpret and advise further.