There are a few private schools that have adopted a “flagship” program. If the kid meets certain GPA and/or score thresholds, they only pay the cost of the flagship school in their home state. The ones I know of are:
Bellarmine (Louisville)
Oglethorpe (Atlanta)
Cornell College (in Iowa, not the University)
I know you mentioned private schools, but University of New Mexico offers very good aid packages as well.