When you die you are judged, that is what we call the private judgment. It is between you and God, and it is immediate. The second judgment is what we call the public judgment, at the end of time, when God will declare for all to know who belongs where.
Imagine when you finish college and 1.) you get that letter in the mail saying you have earned your degree, and then 2.) there is an official graduation ceremony, the handing out of diplomas, etc.
In the first instance, you knew, but in the second instance, the world does.
It is a rough analogy, but that is kind of what it is like. It is not a different judgment, but an official public declaration. Souls in Purgatory are already judged unto salvation [they are saved] and are simply being purified; it is the completion of their justification/sanctification.
There will not be a reversal of anything, all judgments at the first will be the same in the final judgment.