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What's a reliable way for a non-permanent resident alien in the USA to get an auto loan?
You have figured out most of the answers for yourself and there is not much more that can be said. From a lender's viewpoint,  non-immigrant students applying for car loans are not very good risks because they are going to graduate in a short time (maybe less than the loan duration which is typically three years or more) and thus  may well be leaving the country before  the loan is fully paid off. In your case, the issue is exacerbated  by the fact that your OPT status is due to expire in about one year's time. So the issue is not whether you are a citizen, but whether the lender can be reasonably sure that you will be gainfully employed and able to make the loan  payments until the loan is fully paid off. Yes, lenders care about work history and credt scores but they also care (perhaps even care more) about the prospects for steady employment and ability to make the payments until the loan is paid off.  Yes, you plan on  applying for a H1-B visa but that is still in the future and whether the visa status will be adjusted is still a matter with uncertain outcome. Also, these are not matters that can be explained easily in an on-line application, or in a paper application submitted by mail to a distant bank whose name you obtained from some list of "lenders who have a reliable track record of extending  auto loans to non-permanent residents." For this reason, I  suggested in a comment that you consider applying at a credit union,  especially if there is an Employees' Credit Union for those  working for your employer. If you go this route, go talk to a loan officer in person rather than trying to do this on the phone. Similarly, a local bank,and especially one where you currently have an account (hopefully in good standing), is more likely to be willing to work with you.  Failing all this, there is always the auto dealer's own loan offers of financing.  Finally, one possibility that you might  want to consider is whether a one-year lease might work for you instead of an outright purchase, and you can buy a car after your visa issue has been settled.