In searching a few of the other threads, it's clear that a massive Union win at Shiloh would have been hard - Grant was akin tot he star athlete who needs a wake-up call to realize he can't do it all on just talent, he has to prepare a bit, too.
But, presuming something like Lew Wallace choosing to attack when he finds himself behind CSA lines, the Union can find itself with a bigger win, so that noone can say the Buell "saved the day" on the 2nd day - even if Grant probably could have regrouped without him since the CSA's lines were also pretty jumbled. It's still not going to be a huge change, given that Grant hadn't prepared as well as he could have - but just giving the CSA more losses than the Union suffered (let's say reversing the numbers, 10,000 killed, wounded, or missing for the Union, 13,000 for the cSA) would probably prevent Halleck from relieving him and taking over.
But, other than probably getting to Corinth faster, chasing Beauregard down, is there any huge advantage? If Grant takes Corinth a month early, it's going to be a battle, anyway, probably not a siege - perhaps getting those extra 3,000 casualties anyway with the CSA suffering the same. The Confederate troops were already pretty weary; Beauregard might withdraw anyway after some skirmishing. But, the Union would have also been pretty bloddied, and would need time to regroup, get supplies, etc..
Can the Union really get down to vicksburg in 1862? They wouldn't have the added men from Halleck's army, unless Lincoln ordered him there - and it might take a few months to fight their way down there anyway, by which time they wouldn't really have time to combined with the Navy in trying to surround Vicksburg and take it; though the siege begbnning earlier and ending in March of so of 1863 is possible. (OTL the city wouldn't celebrate July 4 for 80 years, maybe here they refuse to allow any Irish to celebrate St. Patrick's Day. )
Onthe other hand, Chattanooga might be easier to take, with an earlier Tullahoma - but the CSA wouldn't be as exhausted, either. Still, if they have Corinth by the end of April, Lincoln might want Grant to come east and take Chattanooga just to relieve the pressure on Washington, though thought being that seeing an army moving East would not only add potential Union supporters to recruit from, it might just cause the CSA to detach some troops from Virginia to send over the try to help defend it, if they can get the rail lines moving and get to Chanttanooga fast enough, say starting in late May. (Yes, that's a long wa, but Lincoln loved playing general and he might think it could work.)
Of course, if Grant is successful in this, it could still be early enough for Grant to be sent East to replace McClellan, so maybe there's an advantage after all. (THough if Grant is in charge of the AotP in late 1862/early 1863, who guards Chattanooga, and who attacks Vicksburg?)
But, presuming something like Lew Wallace choosing to attack when he finds himself behind CSA lines, the Union can find itself with a bigger win, so that noone can say the Buell "saved the day" on the 2nd day - even if Grant probably could have regrouped without him since the CSA's lines were also pretty jumbled. It's still not going to be a huge change, given that Grant hadn't prepared as well as he could have - but just giving the CSA more losses than the Union suffered (let's say reversing the numbers, 10,000 killed, wounded, or missing for the Union, 13,000 for the cSA) would probably prevent Halleck from relieving him and taking over.
But, other than probably getting to Corinth faster, chasing Beauregard down, is there any huge advantage? If Grant takes Corinth a month early, it's going to be a battle, anyway, probably not a siege - perhaps getting those extra 3,000 casualties anyway with the CSA suffering the same. The Confederate troops were already pretty weary; Beauregard might withdraw anyway after some skirmishing. But, the Union would have also been pretty bloddied, and would need time to regroup, get supplies, etc..
Can the Union really get down to vicksburg in 1862? They wouldn't have the added men from Halleck's army, unless Lincoln ordered him there - and it might take a few months to fight their way down there anyway, by which time they wouldn't really have time to combined with the Navy in trying to surround Vicksburg and take it; though the siege begbnning earlier and ending in March of so of 1863 is possible. (OTL the city wouldn't celebrate July 4 for 80 years, maybe here they refuse to allow any Irish to celebrate St. Patrick's Day. )
Onthe other hand, Chattanooga might be easier to take, with an earlier Tullahoma - but the CSA wouldn't be as exhausted, either. Still, if they have Corinth by the end of April, Lincoln might want Grant to come east and take Chattanooga just to relieve the pressure on Washington, though thought being that seeing an army moving East would not only add potential Union supporters to recruit from, it might just cause the CSA to detach some troops from Virginia to send over the try to help defend it, if they can get the rail lines moving and get to Chanttanooga fast enough, say starting in late May. (Yes, that's a long wa, but Lincoln loved playing general and he might think it could work.)
Of course, if Grant is successful in this, it could still be early enough for Grant to be sent East to replace McClellan, so maybe there's an advantage after all. (THough if Grant is in charge of the AotP in late 1862/early 1863, who guards Chattanooga, and who attacks Vicksburg?)