Because my home area is heavily immigrant, Catholic church records are a huge source of information.
Most of us remember the typical: baptism, marriage, and burial. The records should be checked even if you have already located civil records. The Catholic records required sponsors or witnesses, often when civil records do not. These names may turn out to be an unknown sibling, aunt, or uncle.
But you shouldn't forget other sacramental records! Early on, Confirmations did not take place on a regular basis. They occurred only when a member of the church hierarchy visited the area. That meant individuals of all ages were confirmed when St. John's was the only church in the area. Yet, if you are searching for a Connecticut record after 1900, they should appear every year or two at the most. Sacramental records can be used to estimate age when other documents do not exist. I've even used them to estimate marriage records.
Catholic-non-Catholic marriage? In the early 20th century, one individual had to convert or the marriage had to receive a special dispensation from a member of the church hierarchy. That dispensation can provide additional information about the couple's life at the time. In my family's case, it should the applicant's ties to his home diocese - even though he was living across the country
Don't just stop with the baptism records. What you need may be buried deep in another document.