So, the unanswerable question becomes: Would Grant have been able to win a spectacular victory in September/October 1864 that would take the place of the fall of Atlanta, and help Lincoln win re-election in November? The "spectacular victory" would pretty much have to involve either the capture of Richmond OR the severe defeat of Lee's army - either would have been a tall order.
In his own words: No,
and he was actually wanting to abandon the Siege of Petersburg entirely at this time. Without 6th and 19th Corps (Which have been destroyed here), Grant lacked the manpower to go on the offensive and probably would be worse off once Early starts his Valley Campaign. Credit to
@67th Tigers for this:
The 14th July sees the panic ending as far as Grant is concerned. He was been forced into giving up 6th and 19th Corps, and wants them back. Grant's reading of the situation is that the whole operation was a diversion, and Lee will transfer the whole valley force back and overwhelm him, i.e. what Lee tried to do to McClellan in '62.
The 6th & 19th Corps should be got here without any delay so that they may be used before the return of the troops sent into the valley by the enemy. - Grant to Halleck, 14th July 1862
This is Grant's position for the next five months. He believes the Shenandoah is a diversion. On the same day (but sent on the 16th) Grant writes to Sherman. This letter is frank and admits that Grant's attempt to capture Richmond has failed, and in context that Grant believes he can't attack Richmond with the troops he has. He states to Sherman that he intended to make a "desperate effort" to withdraw from Richmond to a threatening position that would allow him to detach troops to Sherman. In Grant's logic he was insufficient force to attack, but the force he has is much greater than needed for defence, thus:
City Point, Va, July, 14th [struck out and 16th written in another hand] 1864
MAJ. GEN. W . T . SHERMAN, CHATTAHOOCHE GA.
The attempted invation of Maryland having failed to give the enemy a firm foothold North they are now returning with possibly 25 000 troops. All the men they have here beyond a sufficiency to hold their strong fortifications will be an element of weakness by eating up their supplies. It is not improbable therefore that you will find in the next fortnight reinforcements in your front to the number indicated above. I advise therefore that if you get Atlanta then you set about destroying railroads as far to the East and South of you as possible. Collect all the stores of the country for your own use and select a point that you can hold until help can be had. I shall make a desperate effort to get a position here which will hold the enemy without the necessity of so many men. If successful I can detach from here for other enterprises looking as much to your assistance as anything else.
U. S. GRANT
Lt. Gen.
The military telegraph passes through the room next to Stanton's office, and Stanton read the transmissions. He copied this and handed it to Lincoln, who was not amused and wrote:
In your despatch of yesterday to Gen. Sherman, I find the following, to wit: 'I shall make a desparate effort to get a position here which will hold the enemy without the necessity of so many men.' Pressed as we are by lapse of time, I am glad to hear you say this; and yet I do hope you may find a way that the effort shall not be desperate in the sense of great loss of life. - Lincoln to Grant, 17th July 1864
This nixed Grant's planned withdrawal. Grant was left in limbo, unable to get the troops he believed were necessary to attack, but probably keenly aware that he was on thin ice and any disaster during the withdrawal would finish him. Hence he was paralysed by indecision and did nothing.
On the 20th, in response to a request for one of the heavy artillery regiments back to man the Washington defences, Grant tries to negotiate an exchange of it for 6th Corps with Halleck. Grant then tries to order it back and is rebuffed, repeating his order on 24th July:
I presume you had not yet received my dispatch directing the 6th Corps to be returned here and the 19th retained.- Grant to Halleck, 24th July 1864
Grant was forced to acquiesce, and it was probably Lincoln who changed his mind. On 31st July Lincoln had a private meeting with Grant. It was apparently, as John Y. Simon observed, not a happy meeting for Grant. It was "an interview without coffee". On 1st August Grant writes not about getting his troops back, but putting his man, Sheridan, in control of them.
Despite Lincoln's correction, Grant's opinion hasn't changed. After laying low for a month by the 29th August he is asking again for 6th Corps:
If this can be done the 6th Corps might be brought here and the 19th left for other service. - Grant to Halleck, 29th August 1864
This is ignored, and the collected papers have no more pleas for 6th Corps until the beginning of October.