I will sometimes just put in a number, then click Civil War (either Union or Confederate) and go down and pick the state that the regiment I'm looking for was from. That should get you rosters as well as the records of the individuals who were in that regiment. That will still probably get you hundreds of guys, but if you click everything except CMSRs, Pension Files and Indexes, that should help you narrow it down. I also happen to know that the 1st CT was the ONLY cavalry regiment from Connecticut, so checking off cavalry on the side should help narrow things down as well.
There is also a regimental history on the 1 CT Cav, but it's modern and may or may not be helpful.
You might also want to check the Connecticut Historical Society in Hartford (they had a lot on the 16 CT guys who ended up at Andersonville) and the New England Civil War Museum, which is in Connecticut. When I was there doing research, I mentioned cavalries to the guy who was acting as docent, and he said there was only one in the state, which makes me think he may know something about the regiment.
Good luck!