Couch was born in Calhoun, a tiny community in Columbia County in southern Arkansas. The eldest of six children, he assisted his parents and younger siblings with the endless work associated with a small cotton farm. His father was also a Methodist minister. When Couch was seventeen, his father's health deteriorated, and the family moved to nearby Magnolia, the county seat of Columbia County. During this time, he was instructed by a future governor of Texas and president of Baylor University, Pat Neff, at Southwestern Academy in Magnolia, also known as Magnolia Academy. He left school early to help with the family finances and assist his mother, the former Marie Heard, with his younger siblings. His first paying job was to start the boiler every morning at Lum Barnett's cotton gin, for which he received 50 cents daily.