Are you looking for something rural, semi-rural, or with lots of shopping? Fairfield is nice, but it's really like a small city to me, nothing like New Canaan/Ridgefield/Wilton. It really depends on what you are looking for.
My IL's bought their home in New Canaan back in the early 1960's. At that time, NC was not considered a desirable location to many, because it was thought of as too rural. They bought their home on five acres for a ridiculously low price, my current car cost more than their home did! People who have lived in NC for that long tend to be very laid-back and down-to-earth, not snobby at all. The town sort of evolved around them. So I'm not sure I would classify NC as entirely "old money", since so many people we know there have "new money", it's a mix of both. Greenwich is more of an "old money" town. NC is a wonderful town to live in, though. Excellent schools, nice parks, the center of town is village-like, and the commute to NY is not bad at all.
Editing to ask, have you considered Westport?
Westport, Connecticut - Wikipedia, the free encyclopedia