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You would have to purchase a brand new vehicle that carried some hefty rebates to not have to put down a large amount of money. I work at a Chevy dealership and we periodically do 20% off the MSRP of new vehicles. Look for something like that to eat up your negative equity because you probably won't get more than 110% LTV from most banks. Definately wait till you your UTI drops to 50% or below and your scores should raise quite a bit.
Oh, and don't go off of KBB. Use NADA for trade values as dealers and banks use them as their guideline.
Just have to watch ads or pull up the websites of the make/model you are looking at and they will have the incentives listed there. If you can find a new 2017 carryover suv right now it should have decent rebates.
All banks are going to limit you on how much negative equity they will roll over. It will ultimately depend on what score, etc they pull when you are ready.
There is really no "best" month to buy a car. You would need to buy new in order to get rebates, yes. The problem with a used vehicle is you will probably be buying it for close to retail value. Then you add tax, fee and your negative and are going to be looking at borrowing thousands over the NADA retail of the vehicle and no bank will do that. You would have to come up with the difference in money down.
If it is a new 2017 Hyundai then it should have rebates right now. This is the best time to buy a new "carryover" 2017 that a dealer hasn't sold yet. Best bet would be to call the dealer or stop in and find out how much under the MSRP you can buy it for. The higher the rebate the less you are going to have to come up with to cover your negative equity.