After the fall of Vicksburg, Halleck wrote Grant that " Before attempting Mobile I think it will be best to clean up a little. Johnston should be disposed of; also Price, Marmaduke, &c., so as to hold the line of the Arkansas River. This will enable us to withdraw troops from Missouri, Vicksburg, and Port Hudson, remodeled so as to be tenable by small garrisons; also assist General Banks in cleaning out Western Louisiana. When these things are accomplished there will be a large available force to operate either on Mobile or Texas. " and "what is to be done with the forces available for the field? This is an important question, which should be carefully considered. If Johnston should unite with Bragg, we may be obliged to send Rosecrans more troops than the NINTH Corps. Some re-enforcements will soon go to banks from the North, but he will probably require troops from you, even after the fall of Port Hudson, to drive Magruder and Taylor from Louisiana. Large forces are comparatively neutralized in Missouri by the forces of Price and Marmaduke threatening the southern frontier of that State. If Little Rock and the line of the Arkansas River were held by us, all of Arkansas north of that river would soon be cleared of the enemy, and all the troops in Missouri, except the militia, could join your army in its operations at the South. If driven from Northern Arkansas and Southern Louisiana, the enemy would probably operate on the Tensas, Washita, and Red Rivers; but, with the gunboats and forces you could send against him, I do not believe he could accomplish anything of importance. If the organized rebel forces could be driven from Arkansas and Louisiana, these States would immediately be restored to the Union. Texas would follow, almost of its own accord." So Halleck had the trans-Mississippi theater on the brain after Vicksburg.Not sure what Halleck was thinking about and who these western generals were. But after the fall of Vicksburg, the trans-Mississippi theater was no longer a critical military one in the fight to defeat the Confederacy. Grant himself had advocated a campaign against Mobile and reluctantly agreed to send a portion of A.J. Smith's Corps of the Army of the Tennessee to Banks, with a strict time proviso after which that force needed to go back to Sherman for operations to the east. Doesn't show much confidence that the RR campaign was all that important.
As for Grant, after he replaced Halleck he sent his first order to Banks which said " I regard the success of your present move as of great importance in reducing the number of troops necessary for protecting the navigation of the Mississippi River. "
The defeat of the Trans-Mississippi confederates on the Red River was expected to allow for a more efficient defense of the Trans-Mississippi (In January 1864 Halleck had written to Steele in Arkansas: " It is hoped that measures may hereafter be concerted between yourself, General Sherman, and General Banks to drive the enemy entirely out of Arkansas and then occupy the line of Red River, which is shorter and probably easier of defense")