It's likely that the dose needs to be lower and not that your thyroid is now over-producing.
I am newly diagnosed with Hashimotos myself, but I believe it is more difficult to manage than hypothyroidism because of the fluctuations in thyroid hormones. My understanding is that it isn't unusual to go from hypo to hyper, at least until the antibodies finally destroy the thyroid's ability to produce hormone which will result in hypothyroid. Your physician will need to adjust your dosage. I do remember a few months ago having hyper-thyroid episodes (storms) where I was having panic attacks, sweating, nausea and light-headeness....before the hypo became the norm. But you know, I will not be surprised in my thyroid swings into a hyper state again. I have read that it can take months, even years to settle down to something stable. I am glad you seem to be tolerating the Levoxyl. I had to switch to a T3/T4 combo a/k/a Armour because of side effects.