I don't know Cozzen's motives but I have to agree with Halleck and Roscran's belief that Union forces could have caught up with and probably destroyed what was left of Van Dorn's forces. If Van Dorn's men could retreat, Rosecran's troops could pursue. It seems from the O.R. Grant was not the originator of "living off the land" tactics, it was Halleck that suggested it to Grant.

"WAR DEPARTMENT,
Washington, October 8, 1862.

Why order a return of our troops? Why not re-enforce Rosecrans
and pursue the enemy into Mississippi, supporting your army on the
country?

H. W. HALLECK,

General-in- Chief.

Major-General GRANT, Jackson, Tenn.

JACKSON, TENN., October 8 1862—7.30 p. m.

An army cannot subsist itself on the country except in forage. They did not start out to follow for more than a few days, and are much worn out, and I have information not only that the enemy have reserves that are on their way to join their retreating columns, but they have fortifications to return to in case of need. The Mobile road is also
open to the enemy to near Rienzi, and Corinth would be exposed by the advance. Although partial success might result from farther pursuit disaster would follow in the end. If you say so, however, it is not too late yet to go on, and I will join the moving column and go to the farthest extent possible. Rosecrans has been reenforced with everything at hand, even at the risk of this road against raids.
U.S. GRANT,
Major- General.

Major-General HALLECK, General-in- Chief."

____________________
David Upton