One thing that I could see happening would be this:
On the night of July 1st, say about 8pm, Lee begins to withdraw his forces back toward Chamberburg. he leaves a small rear guard, perhaps one of Andersen's brigades.
At this time the only union forces near Gettysburg were the remnants of the I and XI corps and some of the III and II under the command of W.S. Hancock.
By chance perhaps, by the time Meade arrives, I believe it was around midnight, the Union is aware that Lee was moving out. Now here is where it gets tricky; either Meade decides to persue Lee and catches him near Chambersburg
OR
Meade decides to pull back to the Pipe Creek Line (
http://www.civilwarhome.com/pcl.htm), in which case Lee would almost have to attack him there. The only realistic way to get to attack Lee in this campaign would be for Lee to slip between Meade and Washington somehow. He would then run the risk of being attacked from the rear by either the Army of the Potomac or the garrison in Washington.
*In regards to the change in Lee's mind:
It's plausible that he has a meeting with Longstreet, Ewell, and Hill, and after some very careful deliberation afterwards with Longstreet, Lee decides that perhaps moving south and getting between Meade and Washington might, after all, be a good idea. This mind change could come about as a result of realizing the strength of the federal line on the night of July 1, and their own disposition of the Army of Northern Virginia. The POD is probably going to be almost insignificant at the time.
The POD cannot include Stuart staying with Lee. It's not the Gettysburg campaign if Stuart is there. I and XI Union Corps don't even try to deploy, much less stand, if Stuart is there. It was Lee's blindness that determined all his decisions on Days 1 & 2. Stuart did not intend to be 60 hours late, but he was planning to "ride around" the enemy. Instead he found himself running THROUGH the AotP, not so easily accomplished, even for his cavalry. As to the cavalry Lee DID have? They were lesser stars, lacking the competence of Stuarts' best subordinates, who he took with him. They could give Lee general information, but not the precise tactical data he had come to rely on from Stuart for Lee's own battlefield decision making.
The Union XII Corps had also arrived by nightfall, and if Slocum had not been his usual "Slow-Cow" self, he could have been in time to save much of XI and I Corps.
One of the problems with the POD is that Lee has reached the point where he believes he and his army had become invincible (following Chancellorsville). Simply showing his men in the open, complete with battleflags flying, was enough to terrify "those people" sufficiently into breaking and running away.
Even if Lee initially heeded Longstreet's advice to maneuver around Meade, it would mean yielding the initiative to the enemy on their own home ground, while allowing Lee to cut himself off from his own supply lines. His men had plentiful food supplies at this point, but ammunition (especially artillery) was another matter. All Meade had to do was lay a light siege against Lee whereever Lee chose to hold his ground (Pipe Creek was not an option for him). With the AotP as the hammer, and the fortifications of Washington as the anvil. Plus the militia of General Couch (20,000 men) arriving, and the 1st division, US VIII Corps also available. If nothing else, Meade could eventually run Lee out of ordnance, while Meade could fight a pitched battle indefinitely.
To make matters worse for Lee, if he HAD chosen to attack Meade at the Pipe Creek Line, it represented a much stronger defense even than the lines at Gettysburg, which was the strongest position the army held as a whole in the war up to that point. No flanking opportunities at all, just headon assaults. And Pipe Creek had an active rail line running directly behind the Union lines, allowing constant resupply and reinforcement.
While Longstreet's criticisms of Lee's attack plans on the 2nd and 3rd Days of Gettysburg were certainly accurate, his own idea for redeployment was actually worse (Assuming Meade chose to hold at Gettysburg). He would have been bringing his divisions of Hood's and McLaws' directly into the teeth of Sedgwick's Union VI Corps (the largest in the AotP). Also, Ewell and his divisional commanders were dead set against a move to the south. As it turns out, they were right. But then, any attack by either side was the worst thing they could have done. Meade and Longstreet knew this. Lee did not.
BTW, ignore Newt's "Gettysburg" trilogy. The first two books are written solely to set up the final novel. The same tired phrase is used in both of the first two books to explain away easy Confederate victory after easy Confederate victory: "HE'S DONE WHAT!?"

In short, one Union general after another defies orders and charges their armies into destruction. Including, of all people, the cautious Major General George Meade.

Nice fiction, but borderline ASB considering some of the personalities involved.
In sum, Lee follows Longstreet's advice and the're not getting home again. Longstreet was too much the tactician to recognize this. Lee understood the greater strategic ramifications of cutting himself loose from his base of supply. Lee was not Sherman, and Pennsylvania/Maryland was not Georgia.