Short answer, it would be October of 1863 before Grant was officially out of the area and in command at Chattanooga, TN.
Although Vicksburg surrendered July 4th, there was still much for him to do, and quite frankly worry about.
There was a very real threat to Grant 40 miles away at Jackson, MS . . . Joe Johnston's
so called "Army of Relief
" .
Bank's predicament in Louisiana also occupied his attention, along with Grant's strategic suggestion to attack Mobile, AL while he had troops in the neighborhood.
As
@wausaubob has pointed out, Grant then visited New Orleans.
I believe he spent about the last half of August & most of September down there.