What is the name of his future brother-in-law?

This may well be his brother, what happened to him after September 1863 is a mystery.

John Maxwell, Private, Company K, 9th Arkansas Infantry, enlisted August 9, 1861 at Camp Lee/October 25, 1861 in Ashley County, Ark./Hamburg, reported on furlough for 36 days from August 6, 1863, remaining absent through November 1, recorded January 1, 1864 as having deserted September 12, 1863, no further records

Note, the next surviving muster roll for Company K, is dated September 1, 1864 at Lonjoy [sp?]., Georgia. Inasmuch as he is not recorded, it can be assumed he had not re-joined the regiment, or had been dropped from the rolls during the intervening period.

M317: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Arkansas

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In his pension application he states he served from Oct. 25, 1861 to the close of war /on or about April 1, 1865. A resident of Crossett P.O., Ashley County, Co. K, 9th Arkansas Infantry, Colonel Dunlop

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Arkansas, Confederate Pension Records, 1891-1935 about J.C. Maxwell
Name: J.C. Maxwell
Residence Location: Ashley, Arkansas
State Served From: Arkansas
Division: Infantry
Company: K
Regiment: Ashley
Vet Application Year: 1910
Comments: approved