You are right, private schools in Houston are likely to be more flexible in their admissions criteria than public schools. Depending on the circumstances and your sons abilities, private schools may give him early addmission. You should also consider the option of magnet and charter schools, which are public school which offer free or low-cost addmissions, but they are less restricted than regular public schools would be.
You can find out more on the Education and Schools in Houston page.
Best of luck