Glenda McWhirter Todd, in her book "First Alabama Cavalry USA: Homage to Patriotism" records these events specific to Company H.
September and October 1863, Company H was stationed at Glendale, Mississippi and November and December, at Camp Davies, MS.
January and February 1864, stationed at Memphis, Tennessee. The Record of Events states the following: "Stationed at Camp Davies Miss, evacuated, barracks destroyed by order of Gen. Stevenson Comdg Dist of Corinth Jan 24/64. Marched to Memphis Tenn reaching this place Sun Jan 31/64."
March and April 1864, stationed at Decatur, Alabama and May through October in Rome, Georgia. (There is no record of events for this company at Rome and no other Muster Roll for them until March 1865, when they were shown stationed at Huntsville, Alabama through June.)
July and August 1865, stationed in Blountsville, Alabama. (This was the last muster roll for Company H.)
Ms. Todd is in the process of publishing a new work (probably in multiple volumes) detailing, as much as possible, each soldier in the Regiment. I believe she is again publishing through Heritage Books in Bowie, Maryland. http://www.heritagebooks.com/http://www.heritagebooks.com/