Assuming your turn isn't over, you'll be able to draw the extra card(s)
were supposed to get during the Draw phase, the same way you draw more
if you increase the Draw amount during your turn.
I take this back. I don't know what I was thinking. This is a Quota-Style mentality which we've since ruled out. During the draw phase you draw as many as you can get and move on.
I've never actually been in a game where this has happened, so I'm just theorizing here anyway. Such a situation is exceedingly rare.