I'll definitely go along with the diagnosis of Hashimoto's, however, I might be a bit skeptical of a hypo dx, at this point... What is the reference range for the Free T4? Reference ranges vary from lab to lab and have to come from your daughter's own report. We need to compare her Free T4 to the reference range to see where she falls within it, in order to determine her actual status.
Your doctor should also have ordered a Free T3 test. Free T3 is another thyroid hormone and is the one that correlates best with symptoms.
As far as the 660 for the antibody count, it's high, but the raw number really doesn't matter that much... the fact that she has antibodies is what counts. There are some people that will tell you that Hashimoto's can be controlled, or even cured, when in fact, it can't be... The antibodies will, eventually destroy her thyroid. The destruction, typically, takes years, but there's no way to tell for sure, as it's different for all of us.
As thyroid tissue is destroyed it will make less and less hormones, so dosage will have to be periodically adjusted to keep up.
What prompted these labs to be performed? Is your daughter having any symptoms at all? If so, please tell us what they are...
Responses don't always post, when we think they do... I always make sure mine post before I move on and if I don't see them, I just type a couple of letters into the next blank space, which seems to "force" out the previous comments that didn't post... that's a glitch that MH is looking into.
Your daughter's FT4 is still at 62% of the range, which actually higher than is, typically, recommended.
Most of us tend to feel best with FT4 at about the mid range point and FT3 in the upper half to upper third of its range. Since we don't know what your daughter's Free T3 is, I might be a bit leery of starting on medication, at this point, since Free T4 must converted to Free T3, which is the hormone that individual cells use.
Have you seen the endo yet? If not, be sure to ask for another set of tests that includes Free T3.
Your daughter will need replacement hormones, at some point, I'm just not convinced that now, is that time, as long as she doesn't have symptoms of hypothyroidism; however, hypothyroidism "can" affect development, so it's important to make sure everything is as it should be.
Why did the doctor originally diagnose Precocious Puberty? It sounds as though her reproductive hormones need to be tested, as well. Has that been done?