$550K isn’t going to go very far in Greenwich, Stamford would be a better option but even there the housing market has gone crazy so your budget may not get you much there either.

For reference, my wife and I bought in Stamford in Jan 2019. 3-BR 2.5 bath, 1700 sq feet for 599k. Out neighbors in a very similar home sold their house last summer for over 700K. There were half a dozen offers on the first day it was listed and the buyer paid cash.