NARA clerks separated 8th Alabama Cavalry files into three groups: Hatch's 8th, Livingston's 8th and just 8th, the last group being records they couldn't readily identify readily as one or the other. These were microfilmed on reels 27 & 28 of the Alabama Confederate CSR's.

If we knew where this Grice person lived in 1860, it might make it easier to determine which command was meant. Here's what I have on Co. "H", Ball's/Hatch's 8th:

Company “H” of Greene, Perry and Marengo Counties

Capt. James Harrison. Formerly a member of Company “D”, 4th Alabama Infantry Regiment. Wounded and captured at LaFayette, Georgia, June 24, 1864. Released at Johnson’s Island, Ohio, June 14, 1865.

On March 4, 1864, officers and men held elections at Tuscaloosa, Alabama, the company letter designation being “I” until April 26, 1864.

On September 23, 1863, Major Lemuel Hatch received authority to enroll exempts for six months service in a local defense battalion. By Christmas three companies had organized at Demopolis, Alabama, other companies still being incomplete. Members (mostly boys under eighteen) reenlisted in Confederate service for three years or the war on February 10, 1864, and went into camp at Tuscaloosa, Alabama.

On April 26, 1864, General Polk appointed Charles P. Ball colonel and Hatch lieutenant colonel. The War Department rejected Polk’s nominations, but Hatch had been taken prisoner by that time, so field and staff appointments were never confirmed. The regiment was usually called the 9th Alabama Cavalry until mid-August of 1864, so some records may be filed under that designation.

Here's what I have on Co. "H", Livingston's 8th:

Company “H” of Walker County

Capt. F. A. Musgrove. Formerly Captain of Company “L”, 28th Alabama Infantry Regiment.

This company was accepted by the State of Alabama on January 12, 1864, and entered Confederate service March 18, 1864. Commissions of company officers bore the former date. After the disaster at Bluff Spring, Florida, March 25, 1865, survivors evidently merged with Company “G” and surrendered with that unit.

This was one of five companies serving under orders of the Conscript Bureau which reported to Livingston's command by early June of 1864. These were added to three companies which had already been in service as Chandler's Battalion. The first three were from the central part of the state; the other five were from north Alabama. Two east Alabama companies of Capt. Sidney A. Moses' Squadron were assigned to Livingston's command but remained on duty at Demopolis and never served with his regiment.