A $2000 1-bed is quite rare in Stamford and, depending on your location preferences, hard to find. Obviously you'll have more luck as you go further from downtown.

Only option is to trawl Apartments.com, Zillow, Facebook Marketplace etc and be prepared to make a very quick decision as soon as something becomes available. I found my place on Apartments.com and put a deposit down 10 mins after seeing it. Good luck!