December 21, 1861 found the 6th Regiment Connecticut Volunteer Infantry (organized in New Haven) camped on Hilton Head. Union forces had captured the island in the Battle of Port Royal on November 7th. It was quickly transformed to serve two purposes. First, it became the military headquarters for the Department of the South. Second, and important to the 6th, it was also the home base for further occupying action in the South. The 6th likely used the time there to rest, train for future conflict, and solidify the Union gains on the island. In protecting the Union advance, the 6th laid the groundwork for the base that was to house a general hospital and eventually thousands of escaped slaves.
There are many ways to research the experience of the 6th. You can start with what are called the CMSR (compiled military service records). Connecticut records are not among those recorded on Fold3.com, so you will have to order the microfilm from the National Archives. Your local library may be able to help you. You can also take a look at The Letters and Journal of a Civil War Surgeon. The compiled memoirs of the 6th surgeon, it should provide details of the 6th's service. Finally, you can order your soldier's pension file from the National Archives. Hopefully one of the sources will have what you need.
For more information, visit the previously referenced sites: