First thing we need to know is the reference ranges for those T3 and T4 results. Second please tell us about any symptoms he has.
First thing to mention is that those results are for Total T3 and Total T4. You should make sure that in the future he is always tested for Free T3 and Free T4. The difference is that the Totals tests reflect all the total amount of serum thyroid hormone. Most of that is bound to protein and thus not biologically active. Only the small portion not bound to protein, thus called Free, is active. If the doctor resists testing for the Frees, just insist and don't take no for an answer. Very important.
From the test results available, even the Total T3 and Total T4 are below the ranges. If his Free T3 and Free T4 show comparable results, that is strong evidence that he is still hypothyroid and needs additional thyroid medication. In fact, the ranges are far too broad due to the erroneous assumptions used to establish them, so even being within the lower half of the range is often inadequate to achieve euthyroidism.
In addition, even though TSH is basically a useless test when already taking thyroid med, his TSH is extremely high.
So, in view of all this, why hasn't the doctor increased his med dosage? A good thyroid doctor will treat a hypothyroid patient clinically, by testing and adjusting Free T4 and Free T3 levels as needed to relieve symptoms. Symptom relief should be all important, not just test results. You can get some good info from reading the first two pages of this link. Note the suggested tests in no. 6 on page 2. The paper could also be used to make sure your doctor is aware of the benefits of clinical treatment.
Further, hypothyroid patients are frequently too low in the ranges for Vitamin D, B12 and ferritin. I highly recommend testing 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.
When there are additional test results, please post them, along with reference ranges, and we will be glad to help interpret and advise further.