You shouldn't be looking for the lowest payment, you should be looking for the lowest overall cost to own whatever you want to own. Don't even let the dealer hear you say that! That's why they love YOU, lol.
If you amortize your current loan, it probably doesnt make sense to pay down your loan unless you can re-fi it and the interest is less over the next __years until it's paid off. You're in the spot where you've payed mostly all interest and now paying more and more principal every month.
Putting big $ down on a new loan would likely be better. Again, run out the costs several ways and see which is for you.