You may procure his records from this message board, at:
http://history-sites.com/research/

These other two Goodwins of Company I must have been kin.

Joseph A. Goodwin

Residence was not listed; 21 years old.
Enlisted on 10/13/1863 as a Private
On 3/16/1864 he mustered into "I" Co. TN 4th Cavalry
He was Mustered Out on 7/12/1865
Promotions:
* Corpl 1/30/1865

.....

Isaac M. Goodwin

Residence was not listed; 20 years old.
Enlisted on 9/10/1863 as a Corporal.
On 3/25/1864 he mustered into "I" Co. TN 4th Cavalry
He was Mustered Out on 7/12/1865

.....

Matthew F. Goodwin

Residence was not listed; 18 years old.
Enlisted on 10/14/1863 as a Private.
On 3/25/1864 he mustered into "I" Co. TN 4th Cavalry
He was Mustered Out on 7/12/1865

Sources used by Historical Data Systems, Inc.:
- Index to Compiled Military Service Records
- Report of Adjutant General State of Tennessee

......................................................................................

Joseph A. Goodwin, Private, Company I, 4th Regiment East Tennessee Cavalry,* USA, joined for duty and enrolled October 13, 1863 at Murfreesboro, Tenn. for 3 years, advanced to Corporal January 30, 1865, mustered out July 12, 1865 at Nashville, Tenn., description on enlistment: 21 year old Farmer, born in Bradley County, Tenn., fair complexion, blue eyes, light hair

* This organization subsequently became Company I, 4th Regiment Tennessee Cavalry

M395: Compiled Service Records of Volunteer Union Soldiers Who Served in Organizations from the State of Tennessee