According to College Navigator (the feds’ website), there are 16 nonprofit private colleges that have acceptance rates of 40% or higher (I had to choose 30% or 40%, 35% was not an option) and that have a population of 10k students or more. Here’s the list: College Navigator - Search Results
Depending on what your price needs are, I’d take a look at DePaul and Loyola Chicago in particular.
If you’re willing to go slightly smaller, then these 15 institutions are private schools meeting the same criteria as above, but that are between 7-10k students: College Navigator - Search Results
There are also some public schools that offer no difference between in-state and out-of-state students. For instance, Central Michigan has a little over 11k undergrads and Grand Valley State has over 19k undergrads. They’re both residential (i.e. not commuter) campuses in Michigan.
We’re also trying to build out this list: Public Schools with No or Minimal Tuition Differences for Out-of-State Students