I'm once went through an issue where an employee resigned, and then tried to un-resign. I wasn't overly disappointed they were leaving--their work was really solid but they were just a drain on me and the rest of the team. If they would have not initially resigned I wouldn't have been at the point where I was letting them go--although I was about 3 months away if I'm telling the truth.
But then they pulled the ripcord and I did have pangs of guilt about taking on their work, which like I mentioned, I really still depended on them for. I could do their work just fine, but there wasn't enough time to do their job AND my job well.
So when they tried to come back, I had a dilemna--first, I'd let the team know and that was kind of embarrassing to think about saying "actually, we're bringing this person back" but then aren't we all adults? Or does one bad apple really upset the apple cart?
It's a tricky situation. My solution was to do 2 things--first, understand where I was falling short in their eyes so that I could course correct. There actually *were* a few things that they had requested which I'd blown past or not considered important or germane to their role--so that was a fair call-out and I needed to improve. The second thing was to let them know what my expectations were for them to return, and their behavior towards the rest of the team was front and center. If they were willing to make an effort to reach out, to try to re-engage the team constructively, and to hit a reset button then I thought it was still worth a shot.
Did I make the right decision? Over the long haul that employee left months later anyway. But I did get some more work out of them that helped us launch some new projects that would have stayed dormant if they'd left the first time. My team actually *wasn't* super happy with me and the decision did undermine my credibility a bit--it gave people lattitude to be a bit more aggressive than they'd previously been with me, knowing that it worked for this other person before. Eventually however, things settled down. The net impact outweighed the embarrassment (such as it was) of reversing a decision.
There's no silver bullet here so sorry if you were expecting a happy ending! And no two situations are the same--eveything is unique. But if nothing else it goes to show the thought you can put into the decision to try to obtain the best outcome for you, for the employee, and for your team.