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I think @sybbie719 is trying to think of /alternative/ ways that you could get aid in order to attend this year.
Yes, your GPA spike is good news, but they already gave you a second year to raise your GPA and you still didn't meet the requirement. You say that it will take you another year of 3.5 each semester to raise your GPA enough to meet said requirement. The committee might not think it's worth the risk.
It's a good idea to look into need-based aid. Failing that, you should consider the possibility of taking a semester off or transferring to a university you can afford if this appeal is denied.
Utah State University - Logan, UT
Application deadline: January 10
Award: Full nonresident tuition for 1 year, full resident tuition for 3 years - encourages students to apply for Utah residency after first year (Presidential Scholarship)
Requirements: See table in link. Lowest accepted GPA is 3.6 with 36 ACT/1600 SAT. Lowest accepted test scores are 32 ACT/1450 SAT with a 4.0 GPA. https://www.usu.edu/admissions/scholarships/freshmen
If you are married, a U.S. veteran, an orphan, an emancipated minor, a homeless youth, a parent who provides more than half of the financial support for a child, or have been in foster care for any stint after the age of 13, you can declare yourself independent on the FAFSA (US News). It doesn't seem like any of these situations apply to you.
You will need to look for merit scholarships only. Full tuition scholarships and full rides do exist, though only for those with stellar stats and not at the most selective or competitive universities. You won't be able to attend any college that is need-based aid only (like Harvard).
I have a chronic illness that sounds very similar to yours, which caused me to miss a lot of school in high school too. I was also looking for a small school with minimal partying atmosphere.
I go to Amherst now, which I agree would be a reach for you, along with Williams and Wellesley (which was also one of my top choices). If you get your SAT up a bit, you'd be in a more comfortable position.
If you're interested in Wellesley, also look into Bryn Mawr and Mount Holyoke, both beautiful schools with great academics but a little easier to get into. Other schools you might like: Allegheny, Susquehanna (has business administration major), Earlham (if you are willing to go a little further from the Northeast), Muhlenberg (previously mentioned).
@ceilingroofgoat Sorry, but that's just not true. If anything, he will have MORE research opportunities with professors (edit: at a LAC), as professors at large universities often use graduate students as research assistants. Amherst and Williams are on par academically with Brown and Dartmouth. I don't know as much about Hamilton, but I assume it's much the same.