I found his pension record which was filed under Company E.

William R. Carroll, Company E, 4th Regiment Arkansas Cavalry, filed for Invalid Class Pension April 13, 1880,
Application No. 358, 061, Certificate No. 273,989.

His widow filed for Widow Class Pension, no date, Application No. 643,021, Certificate No. 446, 885

T289: Organization Index to Pension Files of Veterans Who Served Between 1861 and 1900.