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I am going to retake in October as I didn't do as well as I had hoped in June. But when should I be getting my LOR and applications together to start applying to schools? What happens if I apply to reach schools with my current score before I get my October results? Is there an advantage to applying in September versus waiting till the end of October when I get that score?

Applying around Halloween/Nov 1. when you get an October score back is plenty early, even for ED apps.

You should start getting your LORs together NOW (some recommenders take literally months to get them done), and spend October while you're waiting for your score writing your essays and filling out your apps.