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I'm not sure if a auto loan will do that. I do know when I transferred my installment loan balance to a 0% interest credit card it dropped my score almost 30 points. However I'm saving a ton of interest since my loan interest rate was 29%. I just refianced my auto loan as well so I guess it will drop my score again. I will let you know the outcome onces everything updates.
Yes, you will get credit for having an open installment loan even though the other one closed. It isn't a one for one though, since those equal each other out (relatively), you will still have to contend with the loss of AAoA for the new account and the inquiry.
Another potential factor is that you had paid a percentage of your previous loan (let's say you have paid off 20% of the original loan balance), with the new loan your utilization will be at 100% (this is no where near as bad as having a credit card report over 90%). The short term may see a slight tick downward but I expect it will be a wash before long.