Enrolling or transferring after kindergarten

Has anyone here enrolled their child in public elementary after kindergarten, or transferred schools after kindergarten? If so, did you have any issue getting into your zoned school and were you able to get into schools outside of your zone?

My son will be attending kindergarten in the fall, but we are considering keeping him in a private program only for kindergarten that’s offered at his preschool, then switching him to a public elementary school. We are hoping he can at least get a spot at his zoned school, but we prefer a few schools outside of his zone, but within our district. We are in district 15 and his zoned school is PS 10.