The companies (less Co. K) mustered into Confederate service at Harrisburg, Arkansas, on July 23, 1861. Company K, the "Erin Guards," was from St. Louis, Missouri; the rest of the companies were from northeast Arkansas. The regiment was organized on July 29, 1861, with about 1000 men. By the time of the battle of Shiloh, there were only 306 men able for duty. The regiment sustained nearly 30 percent casualties at Shiloh.
The early history of the 13th Arkansas is discussed in "The Civil War Memoir of Philip Daingerfield Stephenson, D.D.", edited by Nathaniel Cheairs Hughes, Jr., UCA Press [1995]. Stephenson was a member of original Company K, who transferred to the 5th Company Washington Artillery after Shiloh, to join his brother, who was serving in that Louisiana battery.
The 13th Arkansas fought in all the major battles of the Confederate Army of Tennessee, serving in the brigade and later division of the renowned Gen. Patrick R. Cleburne, the "Stonewall Jackson of the West." You can follow the activities of the 13th Arkansas by referring to reports of Cleburne's/Liddell's Brigade and Cleburne's Division. The Official Records are available on-line at several websites.
Due to heavy losses, the 13th Arkansas was field-consolidated with the 15th Arkansas during much of 1863, and with the 5th Arkansas from mid-1863 to the end of the war.
You might be interested to know that Levi H. Tennison subsequently served in the 30th Arkansas Infantry, enlisting in Co. H at Jonesboro, Arkansas, on July 10, 1862. He is carried on the rolls as L. H. Tenison. This regiment was also known as Hart's, McNeill's, Rogan's Regiment and the 5th Trans-Mississippi Regiment.