Records do not indicate the 10th was ever combined with Alabama troops, although they did serve in the same brigade with the 27th Alabama before the surrender at Donelson.
Here's the pertinent part of their history--
From Tennesseeans in the Civil War:
An inspection report November 1, 1861 spoke of the 10th as being in fine condition, saying it was the only regiment at Fort Henry ready for service. In February, 1862, Colonel Heiman was in command of a brigade at Fort Henry, composed of the 27th Alabama, lOth and 48th (Voorhies') Tennessee Infantry Regiments, Culbertson's Battery, and Gantt's Cavalry Battalion.
On February 6, 1862, Fort Henry fell, but all but about 80 men were evacuated prior to the surrender and sent to Fort Donelson, with Colonel Heiman to assume command until the arrival of Brigadier General Gideon J. Pillow. At Fort Donelson, Heiman's Brigade was in Brigadier General Bushrod R. Johnson's Division, comprising the left wing of the Confederate defenses. The brigade consisted of the l0th, 42nd, 48th (Voorhies') and 53rd Tennessee Infantry Regiments, Maney's Tennessee Battery, and the 27th Alabama Infantry Regiment, totaling about 1600 men. Fort Donelson was surrendered February 16, 1862, and the enlisted men in the 10th were sent to Camp Douglas, Illinois.
While there, Colonel James Mulligan, of the Federal Irish Brigade, wrote General Halleck that there were a good many Irishmen in the 10th Tennessee Infantry who wished to take the oath of allegiance and enlist in his forces. Permission to enroll prisoners was denied at this time. A little later, March 19, at Camp Butler, Springfield, Illinois there was published a list of Confederate prisoners who desired to join the Federal forces, but there were only five men from the 10th, so Mulligan seems to have exaggerated.