If you really have hyperthyroidism you were not prescribed the correct medication... Levothyroxine is prescribed for hypothyroidism, which is an underactive thyroid, in which the thyroid does not make enough hormones.
Hyperthyroidism is the opposite; it's an overactive thyroid in which the thyroid produces too much of the thyroid hormones.
Please post your actual thyroid hormone levels so we can see what you were prescribed medication for. Please be sure to include reference ranges as those vary from lab to lab and have to come from your own report.
Are you able to include your thyroid levels in this thread? It would help!
I have Grave's hyperthyroidism, but am on "block and replace" which means that I take Methimazole to block my body from making thyroid hormone and then it is replaced with the correct amount of Levothyroxine. This is not a common approach, so I doubt this is what your doctor is doing, but were you prescribed Methimazole or other anti-thyroid medication in addition to the Levothyroxine?
Just because he is an Endocrinologist does not mean that he is a good thyroid doctor. Many Endos specialize in diabetes, not thyroid. also, most of them have the "Immaculate TSH Belief", by which they only pay attention to TSH, which does not work. If they go beyond TSH they will usually only test for Free T4 and then use "Reference Range Endocrinology", by which they will tell you that a thyroid test that falls anywhere within the reference range is adequate. That is also wrong.
A good thyroid doctor will diagnose and treat clinically, for symptoms first, and thyroid hormones Free T4 and Free T3 second. If you want to confirm what I have said, click on my name and then scroll down to my Journal and read at least the one page Overview of a paper on Diagnosis and Treatment of Hypothyroidism: A Patient's Perspective.
If you would like to try and locate a good thyroid doctor in your area, let us know and we can try to help with that. If you would like further info to read and prepare for the Endo visit, I can also help with that.