You can procure his records through a service provided by these boards, see:
http://history-sites.com/research/
His records are filed in two separate places, noted below.
Jasper C. Thompson, Private, Company D, 6th Regiment Missouri State Military Cavalry, enlisted February 27, 1862 at Knoxville by Kelso for 3 years, left at Calhoun, Mo. in hospital, sick, February, 1863, record in April as absent, gone to Jeff. City as escort for a prison train, absent sick at Milvill, Mo. from September 22, 1863 through October, absent on a scout in Arkansas January 16, 1864, admitted sick, USA General Hospital, Springfield, Mo. July 13, 1864, recorded on furlough from hospital at the end of year, returned to duty January 21, 1865, attending to unserviceable U. S. horses February, 1865, discharged March 22, 1865, description on enlistment: 20 year old Farmer, born in Ray County, Mo., fair complexion, hazel eyes, auburn hair, 5’ 6”
[Civil War Soldiers - Union - MO
… Sixth State Militia, Cavalry, Me-Z AND Seventh Cavalry, A-F › T › Thompson, Jasper C (20), 6 pages]
……….
Jasper C. Thompson, age 20, Private, Captain Kelso’s Company D, Missouri State Militia Cavalry,* joined for duty and enrolled February 27, 1862 at Knoxville for the period, during the war in Missouri, recorded as absent, sick in Calhoun, Mo. hospital at the Jan/Feb, 1863 muster, on furlough at Knoxville, Mo. August 27, 1863, sick at Millville, Mo. September 22, 1863 through the month of October, assigned to permanent Provost Guard, Post of Springfield January 7, 1864,** recorded on the July/Aug, 1864 muster roll as absent, sick in General Hospital, Springfield, Mo. remaining so through December, present at the Jan/Feb, 1865 muster, mustered out at Springfield, Mo. March 21, 1865
* This organization subsequently became Company D, 6th Regiment Missouri State Militia Cavalry
** Card # 22 notes: See James Glenn
[Civil War Soldiers - Union - MO
… Sixth State Militia, Cavalry, Ste-V › T › Thompson, Jasper C (20), 22 pages]
M405: Compiled Service Records of Volunteer Union Soldiers Who Served in Organizations From the State of Missouri