You do have to go to another dealer but its more to get competing offers, what has worked for me is going on the dealers website and picking out the car with #stock number and how many months/miles. Also email the internet manager of the dealer directly, ask them whats the best monthly they could offer you. Theres another good thread on this forum about an internet manager answering questions:
you could also consider re-approaching this dealers internet manage if you didnt start with them. When I was looking for a Mazda CX-5 touring, the base version so 27k~ MSRP, I did get an offer around 270 a month after drive off but didnt end up going for it because it was outside of my budge. I went back and forth with multiple dealers and it wasnt till I came back with my first counter offer from another dealer that I got one of those guys to budge.