Local teacher here: CT has a glut of great private schools. This is partially a reflection on how good our public schools are. My advice is twofold:

1-Decide if you’re interested in boarding. Personally, I wish I had done boarding now that I’ve seen some of what it’s like in a place like CT. Be aware boarding schools in CT are 70k+.

2-If you want a day school, just schedule tours at the 3-4 closest to you that you like the look of. Private schools love giving tours. Since we have so many good private schools, the market drives out the terrible ones for the most part. I teach public in the state, but I’ve mixed in with half a dozen private schools for coaching, PD, friends etc. I don’t really think you need a specific school rec. My advice is decide based on location, price, and campus, and then just tour a few.