I second everything gimel said. I'm another with really low TSH (last one was < 0.03) and I'm a long way from being hyper.
It's still controversial, but studies indicate that you only damage your heart and bones with an excess of FT3, which you weren't even tested for.
Find a doctor who will treat by adjusting FT3 and FT4 to alleviate symptoms and keep you in a range that works for you.
In order to give you the best feedback, we need some additional info. When did you start on the Synthroid and what were the lab results at that time that prompted the doctor to prescribe meds? Since test results are lab dependent, we also need to know the lab's reference ranges for the tests. In addition, total T3 is a somewhat obsolete test. Should have tested for free T3.
In my opinion the very best way to treat a thyroid patient is to test and adjust the actual, biologically active thyroid hormones (FT3 and FT4) with whatever type of medication is necessary to relieve symptoms, without being constrained by resultant TSH level. Obviously your doctor doesn't realize that patients on medication frequently have their TSH level suppressed in order to get symptom relief. This doesn't mean that you are thus hyper. You are hyper only if your FT3 and FT4 level is excessive and you have hyper symptoms. There are many Forum members who have TSH levels below the reference range in order to get symptom relief. My own TSH has been less than .05 for over 25 years.
Bottom line is that you probably need to start looking for a good thyroid doctor that will treat you for symptoms, in the manner I mentioned above. By the way, this doesn't necessarily mean an Endo.