You could have 2 separate quizzes, and the first quiz closes at the deadline, while the second quiz opens at that time. In addition, the 2nd quiz is restricted to be visible only after the deadline. The later quiz has the maximum grade field set to a lower value, but can have the same questions. Both quizzes are placed in the same gradebook category, and the category is set to evaluate to the highest grade of the 2 grade items. Students will see both items in the gradebook, but can only have completed one of them. I believe that if you set the 1st quiz to become invisible after the deadline (using Restrict access), then it will not be visible in the gradebook, but I'm not sure about this.
You may also be able to hide the individual quiz grade items in the gradebook, and just display the category grade instead, to reduce student confusion.