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I was in exactly the same position 20 years ago. My route was to identify employers who were looking for skills sets like mine, rather than specific degrees. In my case, strong leadership skills, evidence of analytical thinking and my outgoing nature got me a graduate entry position with a large logistics company. The best route in for me was Sales, and I took it, then moved into market analysis and then various leadership roles across a number of business units afterwards. In a nutshell, identify companies that value what you bring to the table, then identify roles that don't require immediate logistics qualifications. An easy example would be graduate sales roles at FedEx, UPS, Pitney Bowes or similar - companies where you can grow and move around after proving yourself at foot-in-door stage.
Apply to entry level openings in the Supply Chain. Focus on position with responsibilities that appear less technical and require more coordination and administrative responsibilities. Read up on Supply Chain basics and express your interest and potential in the field during the interviews. Also, if you are system savy that could be a big plus as most SC roles require strong PC skills especially in Excel and Enterprise software like SAP.
I currently have a Sr. Analyst in Supply Chain that started graduated with an English major. She followed the steps above and she is very happy and successful.