Grant likely could have driven the Confederates from the field on the second day even without Buell. The disorganization in Beauregard's command was immense, and the Federal artillery advantage was substantial enough that I think a second day attack would have succeeded. Remember that a lot of stragglers at the river who were separated from their units on the first day in the initial collapse during the night returned to their units or makeshift units that joined the attack on day 2. Grant was getting the better of the fighting towards the end of the day, and on day 2, his forces were more potent.
However, in terms of the strategic level of the campaign, without Buell's troops, meaningful pursuit could not have been given on the road to Corinth (not that Halleck was all that aggressive anyways). Grant could have driven Beauregard from Tennessee, but that was about it. With Buell, a general advance on Corinth was made possible.