Van Dorn and Price retreated to Holly Springs after the Battle of Corinth, where newly exchanged troops reinforced them. Grant then slowly consolidated his position and began moving on Holly Springs, at which point Pemberton retreated across the Tallahatchie.

Facing an entrenched foe across an unfordable body of water, Grant called on help from Sherman in Memphis. Sherman marched swiftly south and crossed the Tallahatchie well to the west of the Confederate defenses, causing Pemberton to retreat to Grenada.

There are not a lot of great sources on this period, and the sources that are available are somewhat flawed. For instance, for all the ink spilled on Van Dorn's Raid on Holly Springs, little notice has been taken of the fact that Lincoln and Halleck had already terminated Grant's move south ("go no farther" were Halleck's exact words).

If you can stand a slow read, the Papers of Ulysses S. Grant for this period are probably the most enlightening source for the federal perspective.