Susiebow, Hashimoto's can make you feel hypo and hyper even though bloods show in normal range. TSH 0.37 doesn't indicate hyper and you will only be hyper if your FT4 and FT3 are over range.
When optimally medicated on T4 it is usually reduced by 25mcg for every 10mcg T3 added to avoid overmedication. As you are sensitive to dose increases it will probably be best to try 5mcg T3 for a week or two before increasing to 10mcg.
Susiebow, T3 may help you tolerate the T4 better but you should still introduce it slowly as I suggested above. Although you aren't optimally medicated, you are taking as much as you can tolerate so your T4 may still need decreasing to compensate for the additional T3.
You don't need to worry about taking T3. Just start with a small dose. What usually happens is that we reduce 50mcg of levo and add 10mcg of T3. Stay at that dose for around 4 to 6 weeks, get a blood test. The blood test will be different from on T4 only. Your T3 will be higher, T4 lower and your TSH low as well. Don't worry about a low TSH as long as your pulse and temp are good. If at anytime you feel overstimulated (hyper symptoms) drop the dose down slightly or miss a dose the following day.
Some of the questions on this page may be helpful to you.
You may be lucky like I was, as soon as I added T3 I immediately felt a big difference in my wellbeing. T3 calmed my palpitations with levo.
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T3 help, maybe adding back NDT
What's the smallest dose of T3
Help - can't tolerate t4 t3 or NDT. 
