What public schools are known for good merit aid for out-of-state students?

Also Ole Miss Academic Excellence scholarship. A 33 ACT basically gives full tuition (~7300 + 12.5k for OOS).
http://finaid.olemiss.edu/scholarships/#8

Wichita State U, mentioned upthread, does give merit scholarships to OOS students. The OOS tuition is about $15,000 a year; I think they give scholarships up to $5000 a year based solely on stats. Now, if you make NMF - check that school out even more - they give a full ride, and if you’re interested in aerospace engineering, it is (hopefully) a good place to go for that. Plus it makes a great safety school admission wise, it’s really easy to apply to and get accepted.

There are also competitive scholarships for those going into engineering that I think go up to $20,000 a year.

Can you explain what you mean?

According to their CDS, Montana State gives very little merit aid to anyone:

In MS, MS State is the ‘known’ engineering school and has good merit for high stats; Ole Miss has come around with more merit for high stats than previous years. Would have to visit both to compare for specifics - I know my kids wouldn’t consider MS State after a visit (college is ‘it’ - rural), but we know plenty that go there and love it. We know kids that attend Ole Miss (for specific other programs) - some with merit and some not.

Be sure to apply to enough of the schools - especially those w/o visits, so you later can visit and make a final selection meeting your criteria.

At Southern Illinois University in Carbondale, your statistics would at least get you in-state tuition, which would make the cost comparable to The Ohio State University; and there is a chance that you would get additional merit aid with your stats.