In all the databases I have access to, there is no T. Cook listed from the counties raised with men from Union County. These are:
3rd (Brazelton's) Cav. Battalion, Company D which became Company E, 1st (Carter's) [Cav.] Regiment
4th (Branner's) Tenn. Cav. Batt., Company C which became Co. D, 2nd (Ashby's) [Cav.] Regiment
5th (McClellan's) Tenn. Cav. Battalion, Co. C which became Co. B, 2nd (Ashb's) [Cav.] Regiment
1st (Roger's) East Tennessee Cav. Regiment, Co. A
[Tennesseans in the Civil War, Part 1]
Part 2, Tennesseans in the Civil War contains a full index of all men serving in Tennessee units. Thomas F. Cook is listed as in the 1st Cavalry, no further identifying information. However, I have not been able to locate a T. Cook in any of the 1st Cavalry's from Tenn.
There is however, a Thomas F. Cook, 1st Georgia Cavalry, see below.
Thomas F. Cook, Private, Captain Reynolds' Company, Lt. Col. Morrison's Battalion, Georgia Cavalry*, enlisted March 22, 1862, at Fair Mount, Gordon County, Georgia by Col. J. J. Morrison for 3 years or the war, deserted October 5, 1863 near Chattanooga, Tenn., no further records
* This company subsequently became Company F, 1st Regiment Georgia Cavalry
M266: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Georgia
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I suggest you obtain his pension application papers, these may be procured through the service noted in the Red enclosed box above. When ordering, you might also ask if they could locate his Compiled Service Records in any of the 1st Tennessee Cavalry units, or any other Tennessee unit. I might note that when transcribing rolls of men, errors always seem to occur and this might explain why I cannot find a record for him in the Footnote CSRs.
His pension papers should identify the correct unit and company in which he served. From there, you should be able to obtain his service records.
Good hunting