Don't start doing it at once. Take the new first draft and shove it onto the backburner. Let it stew back there for a while. A week maybe, or a month. Depends on the writer, so experimentation is probably wise, but revision improves that way.
Well, unless you already know what revisions you need. Then you can revise it immediately -- and shove the second draft to the backburner.
Sometimes it's because the muse keeps on playing with the notions and will come with some brilliant ideas.
But chiefly it's because in the throes of composition, you get too into the story. You need some distance in order to revise. Revising too soon means that you will read what you thought you wrote, what you felt inspired to write, and not the cold hard fact of what you actually did write.
(Beta readers can help here, but I recommend that backburner first. Because once you have burned your beta readers, they will no longer be able to give you an untouched vision of what the story looks like on first reading. They remember that your Rapunzel escaped from the tower on her own at the end, and no longer notice that there is no foreshadowing leading up to it.)