Actually the South did have a "Relief Force" it was turned back by the Union forces near Jackson the Relief Force was Joe Johnston's Army of Tennessee.
As I recall Johnston marched to Jackson and evacuated it after he got there then told Pemberton to link up with his forces ((which didn't happen)) then spent the rest of the time arguing with Davis. However it wasn't the Army of Tennesse because the AOT was with Braxton Bragg at the time and would be until Bragg lost at Chattanooga a couple of months later.