Private Sam Watkins, in his memoir Co. Aytch, had this to say:

Now, those Yankees were whipped, fairly whipped, and according to all the rules of war they ought to have retreated. But they didn’t. Flushed with their victories at Fort Henry and Fort Donelson and the capture of Nashville, and the whole State of Tennessee having fallen into their hands, victory was again to perch upon their banners, for Buell’s army, by forced marches, had come to Grant’s assistance at the eleventh hour. (Watkins, p. 28)

Grant did not follow up on his victory. He did not need to. Having retreated from Shiloh, the Confederates virtually abandoned northern and central Tennessee, leaving the way open for Union invasions from all sorts of angles. Grant also had to fend off his many critics, who blasted him for not entrenching prior to the battle. When a good friend approached Lincoln, asking if Grant should be removed from command, the president replied: “I can’t spare this man. He fights.”