Shiloh was fought on April 6-7, 1862. Pope was fighting at Island No. 10 on the Mississippi River then (Island No. 10 surrendered on April 7). Today, it is about 158 miles by road from New Madrid, Mo (close to what was Island No. 10) and Shiloh -- much further by river as Pope would have had to travel.
In real life, Pope did move to Shiloh with his Army of the Mississippi, but he was not promoted to take over Grant's Army or over Grant. Instead, Grant was kicked up and sideways into a do-nothing second-in-command spot. George Thomas was promoted to Major General and put in charge of Grant's Army. That seems like the most reasonable suggestion for Halleck to do -- exactly what he did in real life.
Two other suggestions:
- William T. Sherman. Sherman was a good friend of Halleck, who had given him cover/protected him when Sherman was in disfavor in 1861. Sherman came out of Shiloh well, with Halleck pushing for him and Sherman being promoted to Major General as of May 1, 1862.
- William Rosecrans. Rosecrans had gotten into a dispute with Stanton, which led to a feud (normal for both, although Rosecrans never seems to have realized his own behavior led to the disputes). That got Rosecrans out of the East and out to Shiloh, where he was appointed to command two divisions in Pope's Army of the Mississippi. Later that year, Rosecrans promotion would be backdated (March 21) to make him senior to Thomas (April 25). Pope's promotion would also be dated from March 21.
Pope, Sherman and Rosecrans would all be available, as would Thomas. All were Brigadiers about to be promoted to Major General when Shiloh was fought. Given what actually happened, Thomas seems the most likely to me. Appointing Pope seems less likely, because it requires someone else to be put in command of Pope's Army of the Mississippi.