The SAO application allows one to apply to a large number of schools with essentially one application, unlike schools using Gateway that have time-intensive customized applications and essay questions. Most boarding schools use SAO, apart from several of those listed above such as PEA, PA, Groton, DA, SPS, Hotchkiss, CRH, L’ville (I may be a little off on this).
Thus you can apply to 10-15 Hidden Gems (in addition to some of the “Gateways” listed above) with virtually one application, but of course you will interview at each one.
Many Hidden Gem schools would welcome your informal inquiry over the summer, and some will be surprisingly upfront about whether it is at all feasible for you to potentially receive financial aid, especially since you seem to be a strong and motivated applicant.