I have just come back from vacation in a country where they call it "winter" but we had sun and temperatures in the 70's(F) almost every day, so my group thought it was summer. Plus the food included many items we think of as only late spring or summer items, such as strawberries, cucumbers, and tomatoes.
If you are shuddering at the thought of another snowstorm and wishing for summer, you can jumpstart the season by joining a summer CSA. Most of their delivery periods start in June or July and run for 10-12 weeks, although one delivering in West Hartford goes for 23 weeks! Some of the more popular ones in the Hartford area include Urban Oaks, which is almost full for the summer season, and Holcomb Farm, which includes an option to deliver pre-packed boxes to two or three sites in West Hartford.
If you prefer to pick your own veggies, my CSA allows that. The farmers do the actual picking but instead of prepacking they deliver crates of herbs, fruits, and veggies and you get to choose the ones you want. This can be an advantage if your family really likes cauliflower - or if they don't like it at all - as you can select a head to meet your needs. They even offer a "Swap Box" where you can deposit items that your family refuses to eat. At the end of the day, all the leftovers, including anything not claimed in the Swap Box, go to local food pantries or the kitchen at Marshall House.
If none of these are convenient to you, the Connecticut NOFA website includes a list of CSAs by county, so you should be able to find one near you.