I have slight hypo problems right now. I'm not an expert, at all, as I've not found anybody willing to dig deeper into it. According to a couple of doctors I've seen, there are several reasons the thyroid could not be functioning correctly without you necessarily having an autoimmune disorder. The biggie, apparently, is adrenal fatigue. The hormone production of the adrenals is intimately connected to production of T3 and T4 - or maybe just one of the two. I don't remember the exact reasons right now, but if you look it up you should find good information. I have pretty severe adrenal fatigue - another thing I have not successfully taken care of. Theoretically, if your adrenals function properly again, it will help the thyroid to function properly, as well.
An iodine deficiency can also cause hypo problems. If you've been avoiding iodized salt like the plague, that could have starved the thyroid of this essential nutrient. Apparently, the thyroid is the only part of our bodies that effectively absorb and use iodine. There are ways you can get iodine outside of table salt, which I don't believe is healthy anyway. Kelp is the most highly recommended source. I haven't done that yet simply because iodine is also supposed to be very unhealthy for people with severe acne, like me. Not sure what to do about that.
The pituitary gland also in essence controls the amount of hormone the thyroid produces. It produces TSH, which signals the thyroid to produce its own thing. When blood levels of T3 and T4 are down, the pituitary produces TSH in response, which stimulates the thyroid to get its act together and produce more. If the pituitary gland is not functioning properly, this can cause the problem. I have a very messed up pituitary gland, as well (I have PCOS, so it's spitting out hormones in wacky amounts, some too much, some too little). Once again, sadly, it's hard to get this back in balance, and I haven't been successful. The hypothalamus controls the pituitary gland, as well, so if that gland isn't functioning well, everything else in the chain of command won't be working properly.
I'm sure there are a million more reasons why the thyroid could be behaving strangely, but these are the ones that came up in my particular case. I wish I had better information for you regarding solutions, but I haven't found one yet. I am slowly improving my adrenal function through supplementation and (up until the last few weeks) avoiding stress. There are adrenal stress index tests you can take to show the output of the adrenals. The pituitary gland is apparently the most mysterious problem and nearly impossible to "fix."
I haven't gone on meds yet, because it hasn't been bad enough. I'm hoping to balance things out naturally. I'll do meds if what I'm doing doesn't work and things get worse.
I wouldn't worry that Hashimotos is the only source of your problem. There are plenty of other reasons out there. Good luck!