I have been unable to locate him in the 37th Georgia or any other Georgia unit following the break up of the 3rd Battalion
Joseph W. Stephenson, Private, Company C, Battalion of Artillery, Georgia Volunteers,* enlisted June 25, 1861 in Cobb County, Ga. by Capt. Kendrick for the war, last recorded present on the end of year muster roll dated January 14, 1863 at Shelbyville, Tenn., no other records
* This company subsequently became Company C, 3rd Battalion Georgia Infantry. The regiment was broken up May, 1863, Company C consolidated with others to form the 37th Regiment Georgia Infantry May 6, 1863
M266: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Georgia
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Troup County, Georgia
3rd Battalion Infantry Georgia
The 3rd Infantry Battalion was organized at the Upson Camp Ground, near The Rock, Georgia, during the summer of 1861. It was formed with three companies, later increased to five. The unit took part in the Cumberland Gap and Kentucky Campaigns, then serving under General Rains lost thirteen percent of the 300 engaged at Murfreesboro. In the spring of 1863 the battalion merged into the 37th Georgia Infantry Regiment. Lieutenant Colonels Anthony F. Rudler and Marcellus A. Stovall were in command.
http://www.rootsweb.ancestry.com/%7Egatroup2/3rdbattalioninfantry1.htm
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The Georgia 37th Infantry Regiment was organized during the spring of 1863 by consolidating the 3rd and 9th Georgia Infantry Battalions. Many of its members were from Murray, Jackson, Franklin, Elbert, and Hall counties. The unit was assigned to General Bate's, Tyler's, and J.A. Smith's Brigade, Army of Tennessee. It fought with the army from Chickamauga to Atlanta, endured Hood's winter campaign, and was active in North Carolina. The 37th lost fifty percent of the 391 engaged at Chickamauga and in December, 1863, totalled 416 men and 265 arms. Few surrendered on April 26, 1865.
FIELD OFFICERS:
Colonel A.F. Rudler; Lieutenant Colonel Joseph T. Smith; and Majors Jesse J. Bradford, Meredith Kendrick, and R.E. Wilson.
http://www.researchonline.net/gacw/unit88.htm
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The 37th Regiment of Georgia Volunteers, Tyler's Brigade; Bate's Division, Cheatham's Corps., Army of Tennessee; organized May 6, 1863, from the 3rd and 9th Georgia Battalions