Affordable, high-quality private school accepting kids mid-year? Does this unicorn exist?

My kid’s elementary school is in a downward trajectory this year (thanks to the state takeover of HISD). I’m short on options but don’t want to procrastinate and have her miss any more years of education, after the pandemic really gave her a lousy start.

Are there any good private schools fairly central? My older kid is in an ok public school and I want to stay close for easier logistics.

Hoping for non-religious, but would take a liberal religious school in a pinch.

Location, tuition and ability to get in are the biggest concerns.

My kid has the GT label and high grades, if that matters for admission.