I love the comedy scene down here! If you move between New Haven and lower Fairfield County (like Stamford) there are a few clubs:

• Fairfield Comedy Club (Fairfield, CT. Smallest club, lots of local comedians open for headliners who are usually smaller-name NYC circuit comedians)
• Stress Factory (Bridgeport, CT. Most established, gets big names)
• New York Comedy Club Stamford (Stamford, CT. Brand new club! Most of the support from the show I went to were local comics)
• College Street Music Hall (New Haven, CT. Not really a club, but it’s a venue that gets bigger-name comedians often enough!)

In Hartford they have the Funny Bone, but I’ve never been/not heard too much about it. You’ll have better options moving to New Haven or Fairfield County because of your proximity to NYC. Good luck!!