On April 8 Grant sent out a small pursuit party under Sherman to see what the Confederates were up to. Sherman was repulsed at Fallen Timbers. This ended the Union pursuit. As stated earlier the purpose of Grant’s army being at Pittsburg Landing was be in a position to capture Corinth. Both Grant and Beauregard’s armies had suffered severely at Shiloh; casualties on a scale that the country had never experienced before. On April 11 Halleck arrived to take command of the combined Union force. Pope's army would be brought over to Pittsburg Landing after its capture of Island No. 10. After April 11 Halleck had a relatively fresh army under Buell to tip the odds in his favor. An immediate advance, by a general who was willing to attack the Confederates at Corinth, probably would have resulted in a major victory. Halleck was too worried about long casualty lists to risk an attack that might bring a major victory but also a large casualty count.
Grant’s other options were to have used many more men in the pursuit, Buell’s Army of the Ohio had not been too bloodied on April 7. Also Grant could have decided after Sherman returned to make a larger pursuit on April 9. Though it is well after the battle of Shiloh it is possible that Halleck could have ordered a pursuit of the Confederates soon after his arrival on April 11.
Grant mainly didn’t pursue because he didn’t feel he could order Buell to do it and because he knew that Halleck wanted to avoid another battle. Halleck was on his way to Pittsburg Landing to take overall command. If Grant had made a large scale pursuit on April 8 or 9 he probably would have captured large numbers of Confederates and that might have softened the news of his own large casualties in the battle. There is also a possibility that if Grant had captured a large number of Confederates on April 8 it may have convinced Halleck to start the Corinth campaign sooner, move faster and attack. Halleck might have been able to score a major victory at Corinth that included the capture of a portion of Beauregard’s army as well as the strategic rail junction. Considering how cautious Halleck was it is unlikely that anything would have made him move faster and attack.
The Confederates would have been able to rest somewhat before Halleck's army arrived. There were also fortifications around Corinth that were made stronger every day. Halleck had a superiority of numbers though and probably could have taken Corinth with light to moderate losses.
It is a critical decision to not pursue. There were very good reasons not to pursue but there were also good reasons to pursue. Every day the Confederates have to strengthen Corinth means it will take m ore men to capture it. Halleck wants to avoid casualties and rather than rush into a risky battle he chooses to march slowly and continuously entrench so that he cannot be attacked.