What is your current rate? What CR changed? And how sure are you that a lender would use the same report and score? Did the dealer in August provide you with a score?
I don't know the answer to whether or not you can refinance. I don't see why not. You may want to look at your agreement to be sure to make sure.
Ignore "mortgages" in the link above; you can use these amortization tables for a car loan too. I don't know your rate, but even decreasing from 19% to 12% can save you about $100/mo, despite only paying about $700 in principal over the prev. 3 months, based on an example of $23k borrowed at 5 yrs.
A 623 FICO score, assuming this came from myFICO, isn't all that high. Better than 588, though. Congrats. I bet with a month's worth of effort with repair, you can bring your FICO scores into the high 600s and have that in place by January and potentially qualify yourself for a single-digit rate.
Do you belong to a CU? Have you spoken to one? That is the nice part of CU's is you can talk to a live person, explain circumstances and goals. CU's are there to help members improve their finances through better loan rates and better rates of return....if you are qualified.
Also, score alone doesn't dictate approval and rate....the actual contents, your income and debt are important considerations as well. The higher the "risk" the higher the rate.