“Next tier down” is kind of a vague concept as rankings are so subjective. I our family’s case, my daughter wanted to do engineering. It would have been quite a long shot, as it is for every applicant, to go to MIT. She was interested in Worcester Polytechnic Institute (WPI) which I consider about that one tier down level. She was accepted with a fair amount of merit aid. We do not qualify for any need-based aid. It wasn’t a full ride but made it quite affordable especially compared to other colleges she was looking at.
WPI was looking to increase the number of women in the college to get a better balance between the sexes. I believe that helped her get a little more money to entice her to attend.
Most, if not all, colleges publish a document called the “Common Data Set”. Google that with the college’s name and you can find it. Section H describes that college’s financial aid picture. They break out the “need-based” and the “non-need-based” aid. The top tier colleges will have all, or almost all, their aid in the need-based column. So, you can do the research and find out how any particular college hands out their aid.