Given Grants habit of throwing hundreds of men against his opponants, even when there are well constructed defenses before him, then it is not totally implausable that, had Grant face Lee at any time before Gettysburg in Virginia, he would have been defeated soundly by Lee and lost his standing in the Union Army and the history books completely.
If, for example, Grant was in charge of the AOTP at Fredericksburg then he would have been defeated with relative ease by the ANV and would have had his reputation so severly damaged that it would be unlikely that he would ever hold a high command post again.
However if Grant had been in charge of the AOTP during Antietam then it would be far more likely that he would overwealm Lee's forces and either drive them back with heavy casualties or defeat them completely.
Also it is worth remembering that Pope was an offensive general brought east early in the war and he was defeated by the vastly superior Confederate Generals Lee, Jackson and Longstreet. At this stage in his career Grant was no where near good enough to get the better of Lee and his army in the east.
The best option out of the two given would cirtainly be for Grant to march into the Deep South and march through Georgia, the key to the Confederacy, earlier in the War. However with there still being relatively large forces being able to oppose him, remembering that Beauregard would not have lost Corinth if Grant was moved east and Bragg would not have invaded Kentucky, then it is unlikely that Grant would have been able to march straight through the Deep South at this proposed point in time.