My great-Great uncle, Andrew Jackson Caldwell from Franklin Co., also served in the 3rd Tennessee, approximately the same dates as your gGrandfather. I haven't been able to find many details about the activities of the 3rd TN except that the regiment was formed too late to participate in the actual fighting. A Web reference for this is: http://tennesseeencyclopedia.net/imagegallery.php?EntryID=M093http://tennesseeencyclopedia.net/imagegallery.php?EntryID=M093
There also is a book entitled "Volunteers: Tennesseans in the War With Mexico". Kitchen Table Press. 1986, by Brock et al. which, however, I have been unable to find at any of the used-book sites as yet (it would surely be in the LoC).