3.5UW, in the recent past, has been enough to qualify automatically for rather large merit scholarships at a few schools.
http://automaticfulltuition.yolasite.com/
Whether private or public schools will offer more depends on your qualifications and the market you’re considering. Check Kiplinger’s, the online NPCs, and the CDS section H for any that interest you. Or keep it simple and just focus on in-state public schools, possibly including ones within commuting distance.