Not all subacute thyroiditis goes away... sometimes it's Hashimoto's, which is autoimmune and is permanent. Have you been tested for thyroid antibodies to determine if you have Hashimoto's? Your symptoms (hyper, hypo, hyper, etc) sound exactly like beginning stages of Hashimoto's.
The antibody tests you need are Thyroid Peroxidase Antibodies (TPOab) and Thyroglobulin Antibodies (TgAb). You need them both because they're both markers for Hashimoto's and some of us have one or the other and some have them both.
Just because labs are "in range" doesn't mean they're adequate; both your FT4 andFT3 are low in the ranges. Most of us tend to feel best with Free T4 in the mid-range and Free T3 in the upper half to upper third of its range. As you can see, your FT 4 is on the floor of the range and your FT3 is only at 30% of its range.
Some symptoms can apply to both/either hyper and hypo. The feverish feeling could be because you have inflammation and simply feel "ill". IMO - yes, you need a higher dose of medication.
There are other possible scenarios as well...
There's no time-frame for subacute thyroiditis to resolve; it can take a year or more and if you were diagnosed in June, you could have a ways to go yet. It may still go away on its own, even several years from now.
There's also no time-frame for subacute thyroiditis to convert to Hashimoto's; that can happen at any time - also several years from now or even after the subacute thyroiditis has seemingly resolved.
A thyroid ultrasound would be in order, as some people can be diagnosed with Hashimoto's, based on thyroid characteristics, even with absent antibodies. It's possible the antibodies simply haven't ramped up yet. An ultrasound would also determine if you have nodules, which are common with Hashimoto's.
The important thing will be to maintain adequate thyroid hormone levels to alleviate symptoms while you have thyroiditis. You should also ask your doctor to continue to test antibodies periodically, so if they become elevated and your subacute does turn into Hashimoto's, you'll know it.
In the meantime, things you can do to help yourself are to make sure your Vitamin B-12, Vitamin D, and Ferritin are all in optimal ranges since these vitamins/minerals are all necessary for the proper metabolism of thyroid hormones. If you haven't had these tested, I'd recommend you do so. Feel free to post the results and reference ranges, here so we can help interpret them, since "in range" isn't the same as optimal. Selenium is also necessary for proper metabolism of thyroid hormones and aids in the conversion of Free T4 to the usable Free T3, plus it's been shown to help reduce thyroid antibodies.