Korteweg-de Vries (KdV)-type models are frequently seen during the investigations on the optical fibers, cosmic plasmas, planetary oceans and atmospheres. In this paper, for an extended three-coupled KdV system, noncharacteristic movable singular manifold and symbolic computation help us bring about four sets of the auto-Bäcklund transformations with some solitons. All of our results rely on the coefficients in that system.