You shouldn't need to take the DAT again. LORs should be strong, so when you ask for someone to write you a reference letter, ask "Can you write me an outstanding letter?". If they can't move on and find someone who will.
Most schools have already filled 90% of their class by the time they got your DAT score. And then they'll fill the other 10% off the people on the wait-list, or 'outstanding' applicants (3.8+, 23+AA) who applied late.
General rule is to apply before September 1st for serious consideration. The earlier the better though. You're golden when the new application cycle opens up on June 4th. Good luck.