I think, for the purposes of this thread, we can consider “upper middle class” to mean “let’s just postulate merit aid only.”
Rice does give some merit aid; Duke and Brown do not. For schools like Rice, merit offers are discretionary, so you can’t know how much they may or may not offer you until you get your admissions decision. (But you can know which schools definitely do NOT offer merit, such as all the Ivies, Duke, Stanford, Pomona, and many other elite schools.) Vanderbilt and WashU are others in Rice’s tier that are relatively generous with merit. UChicago gives some merit too, as does Claremont McKenna.
There are automatic-merit schools where you could attend essentially free with your stats. That 36 ACT and 4.0+ GPA will get you a full ride offer from U of Alabama, and a great Honors College education with a majority out-of-state student population, many of whom would be there for the “big merit” just like you. It’s something to think about. UA has gotten a lot less generous with merit now that their Honors College is well-established as a magnet for high-stats OOS students; but they still give the full-ride deal for a 36/1600.
USC and Northeastern are particularly generous with NMF. Most other schools at this level of competitiveness just boost the merit aid you would otherwise have gotten by a few thousand dollars for NMF.
A number of the women’s colleges, if you’re female, give good merit scholarships (not Barnard or Wellesley, but Scripps, Smith, Bryn Mawr and especially Mt. Holyoke). Other top-tier schools that give good merit include Case Western Reserve U, U of Rochester, U of Richmond, and Tulane. Excellent publics with good merit scholarships include Pitt and Michigan State. Also some of the more competitive OOS universities in the WUE program would be extremely affordable for you, between the WUE discount and the additional merit you would receive. (U of Utah, U of NM, UNLV/UNR, Washington State, Colorado State.)
If you go down a level of competitiveness from the above, then you can get even bigger merit awards from many schools. But my guess from your post is that your baseline is UC, probably with Regents, and that you’re mainly interested in schools of UC-level competitiveness and above. So hopefully the above addresses what you were asking.