When the Union began deploying at Pittsburg Landing C.F. Smith was in charge of the operation. He chose Savannah as his headquarters. When Grant replaced him he continued to keep the army headquarters at Savannah. He spent his days at the landing so Savannah really served only as a comfortable place to stay. In early April word had been received that McClernand was being promoted and would be the ranking general at the landing, Sherman was the ranking general when Grant was not at the landing. Grant had decided to transfer his headquarters to the landing so that McClernand would not be in command. If the Confederates had attacked a few days later Grant would have been at the landing.
Grant could have moved his headquarters to the landing anytime but he chose to remain at Savannah. There is a wide held belief that Grant kept his headquarters at Savannah so Sherman would regain his confidence by being the man in charge at the landing.
Grant's influence on the battle if he had been at the landing from the beginning is debatable. There is a chance that the army would have been better prepared for the Confederate attack. Sherman did not tell Grant of every alarm his colonels reported but if Grant was at the landing he may have been better informed of the local situation.
There is probably very little Grant could have done to effect the course of the battle if he had awoken at the landing so this is not a critical decision.
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