On Fold3.com, I found at least two William Coles in Alabama. Based on the letter you describe being from Tennessee in July 1862 or 3 (I'm thinking 1862), I'd say it's the first man.
First packet: Eight pages in a file for the 50th infantry, Alabama:
Page 1: cover page with his name and the enclosed card numbers (subsequent pages)
Page 2: William Cole, pvt., Capt. Haney's Co., Alabama Volunteers, age 32.
Mustered in at Camp Jordan, near Huntsville, Alabama, Oct 4, 1861
Joined for duty and enrolled Aug 4, 1861 in Florence, Alabama by Capt. Haney for a period of 3 years.
Travelling ... 75 miles.
Page 3: Wm. Cole, corp'l., Company D, Battalion Alabama volunteers, Company muster roll, September 27 to Oct 31, 1861
Remarks: promoted from private to Corporal 24 Oct 1861.
Page 4: Wm. Cole, Corporal, Co. B, 2 Battalion, Alabama Vols., Company muster role for Nov & Dec 1861. Present.
Page 5: Wm. Cole, Corp., Co. D, Davis' Independent Batt'n, Alabama Vols., appears on Regimental return, November 23, 1861. Absent men accounted for: Absent with leave at his home in Lauderdale Co.
Page 6: W. Cole, Pvt [!], Co ___, 26th Reg't Alabama Infantry. Regimental return for the month of Jun 1862. Absent men accounted for: Sick, sent to Hospital at Columbus [or?] Enter ferrie? Miss[issippi]
Page 7: Wm Cole, Corp., Co. W?, or N?, 26th Reg't Alabama Infantry for the month of July 1862. Alterations since last return. May 26th 186_, Corinth, Miss., Deserted
Page 8 (typed): C | 26 | Alabama. Wm. Cole, Corp., Co. H, Appears on a consolidated report of deserters in the Reserve Division, First Corps, Army of Mississippi, for the last eight months. Report dated, Tullahoma, Tenn., Nov 1862. Brigade: 1; Residence: Lauderdale, Alabama
Second packet six pages: Page 1: Cole, William, Co. C., 4 Alabama Vol. Mil. Corporal.
Page 2: Wm. Cole, Corp'l, Capt. Howze's Company, Wyatt Guards. Appears on a muster role attached to the 2d Regiment Alabama Vol. Militia. for March 15 to April 30, 1862. Station Fort Gaines. Enlisted March 15, 1862; in Marion, Alabama by Jno. Howze for a period of 90 days. Present.
Page 3: William Cole, 2 Corpl. Co. C, 4 Reg't Alabama Volunteer Militia. Appears on Company Muster Roll for March 15 to June 13, 1862 [repeats enlistment info from prior page]. Present.
Page 4: William Cole, Corpl. Co. C., Byrd's Ala. 90 Days Troops. Date of discharged June 13, 1862; day of payment June 13, 1862 by G. W. Holt.
Page 5: Payment Voucher. $39.00
Page 6: Form no. 4; Certificate to be given soldier at time of his discharge. I certify that the within named Wm. Cole a 2d Corpl., of Capt. Howze, Company C., of the 4th Regiment, of N [A?] .V.M. born in Perry Co., Ala. in the state of Ala. age 35 years, 5 feet 5 inches high, light complexion, blue eyes, light hair, and by profession a doctor, was enlisted by Capt. John Hoyze, at Marion, Ala., on the 15th day of March, eighteen hundred and sixty 2 to serve for 90 days and is now entitled to a discharge by reason of the expiration of the term of his enlistment. The said Wm. Cole was paid by Paymaster at this time... pay due from March 15th to June 13th 1862 inclusive.... $39... signatures...