There is a James A. Brooks on the roll of the Union 4th TN Mounted Infantry Co.A. They were organized 9/1/64, mustered out at Nashville 8/25/65. Might explain them being in Illinois.

Fourth Mounted Infantry
TENNESSEE
(1-YEAR)

Fourth Mounted Infantry. -- Lieut.-Col., Joseph H. Blackburn;
Maj., Thomas Waters.

The recruiting of this regiment was begun in Aug., 1864, and
the last company was mustered into service in February
following.

Its members were principally from the eastern portion of
Middle Tennessee. It was placed under the command of Joseph
H. Blackburn who was commissioned lieutenant-colonel Nov. 26,
1864.

Co. B was organized at Memphis as a detachment of Union guards
from Jan. 1 to April 14, 1864, to serve for one year. It was
mustered out June 1, 1865.

The remaining companies were mustered out Aug. 25, 1865, in
accordance with orders from the war department.

Source: The Union Army, vol. 4, p. 389