I read about someone finding their Civil War ancestor’s letters in a trunk in their attic. Sigh, I wish that would happen to me. Unfortunately, I have no attic with dusty trunks waiting to be opened.
I have some World War I letters from my grandfather and a great-uncle. They were treasured and kept carefully by my mother, our family historian. If you have someone in your family, who serves in that role, check with them.
Some historical societies have letters donated by local residents. These are not always digitized or displayed. They may be cataloged and stored away. Ask about Civil War letters and diaries in historical societies in the area where your ancestor lived.
Even if what they have is not your ancestor, it is possible that your ancestor was mentioned by another soldier. Reading about other soldiers in the same unit as your ancestor will give you details about their Civil War experience.
Check eBay, as Civil War letters sometimes show up on there. You can place a watch on keywords or a topic and eBay will inform you when something shows up at auction.
Many letters from that era have been published in books. Search for Civil War letters on Amazon under the book category. Some books focus on letters by an individual but others are collections of letters by various soldiers or a military unit or a region.