Unfortunately, there is very little on Lt. Spencer

John L. Spencer, 2nd Lieutenant, Captain Alexis T. Rainey’s Company Texas Infantry,* enlisted June 24, 1861 at New Orleans by Lt. Jno. G. Deverau for the war, receipted $420 , Commutation of rations for 56 men on April 16, 1862 at New Orleans, La., killed May 1862, no further information

* This company was successively designated as Captain Rainey’s Company, Texas Infantry; Captain Gaston’s Company and Company H, 1st Regiment Texas Infantry

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Company was organized in Anderson County, Texas, and was mustered into Confederate service "for the war" at New Orleans, June 24, 1861 by Lt. John G. Deveroux, CSA.

May 7, 1862

Attacked Yankees at Elthem’s Landing and drove them back into the river. At ten o'clock Whiting's forces took up the march as the rear most Confederate division with the Texas Brigade assuming the function of the rear guard of the entire Southern army. That night, Whiting ordered Hood and Hampton to lead their brigades back to their bivouac area north of Barnhamsville and remain there until the trains had cleared the road to Richmond. Lieut Col. Harvey H Black is Killed.
http://texas-brigade.org/1st_tex/1tex.htm

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SPENCER, JOHN L. 2Lt. - Detailed to recruiting serv. in Tex., Feb.-Apr., 1862: K., Eltbam's Landing (May 7, 1862), and buried in Hollywood Cemetery, Richmond.
http://texas-brigade.org/1st_tex/1texrostercoh.htm