If your ancestor served in the Connecticut regiments during the American Civil War, you might have missed a record source. Fitch's Home for Soldiers served as the late 19th centuries equivalent of the modern veterans home. Soldiers were able to live there during the end of their lives, and receive care. Pensions were paid to them at the home, they may have been buried from the home, and they had prove their service. The end result - a rich paper trail.
So how do you access these files? According to a Connecticut State Library webpage, access is onsite at the state library. On the plus side, an online database allows you to tell if your ancestor lived in the home. You can search by name, unit, or town. And if you need only your ancestor's regiment, this may be another place to check.