To answer your question, it depends on the school. There are schools where it is much much easier to get in as a day student because they are in remote areas with a limited pool and accepting local kids is one of the easier ways to foster good community relationships.
For others, where there is a strong local applicant pool and appetite for private school and a fixed number of seats for day students, it could be harder. Ten years ago, the folks in admissions at Lawrenceville would tell you that you could improve your odds by applying as a boarder. (We heard what @sgopal2 did.) I’m not entirely surprised that this has changed as local demographics have started to shift as well, both changing the size of the pool as well as some of the factors that made LPS less attractive.
Someone looking at Hill and Perkiomen as day options probably cannot consider most of those others suggested here as day options. Boarding, of course! Westtown might be a local option depending on where you are coming from.
At this point, you’ve thrown your hat in the ring for those two. I don’t know how different admissions are for day and boarding at them. At any school, they aren’t going to take unqualified kids from either pool to fill the class.
If he doesn’t get in as a day student, you can always inquire as to whether switching to boarding would give him a place. At most schools, though, the cost differential between the two can be significant.