I don't know of any historical work specifically about Co. H of this unit, but there is a capsule history of the unit at:
The only other sources I know of for this unit are:
(1) "The 19th Mississippi Regiment in the Civil War (With Notes about Co. E and Pvt. William Meek Furr," by William Frazier Furr. Montgomery, Alabama. Privately printed: 1992. A copy of this work may be in the MS Dept. of Archives & History; hard to tell from the citation I found in Dornbusch, "Military Bibliography of the Civil War," IV.
(2) "Movements of the Confederate Army in Virginia and the Part Taken therein by the Nineteenth Mississippi Regiment, from the Diary of Gen. Nat. H. Harris," compiled and published by Capt. W.M. Harris. Ducansby, MS: 1901. 45 pp. A copy of this work may be in the Maryland Historical Society, as one unattested abbreviation after this entry in Dornbusch was "MdHi." Similarly, a copy may be in the Virginia archives, as another unattested abbreviation after this entry was "Vi." A third unattested abbreviation after this entry was "CU," which I have no way of deciphering.
Wish I could have been of more help! -- Jim Huffman, Gainesville Vols