I have not looked into this recently but when I was doing some serious library research I was able to purchase a university library card even though I was not a student. One university just gave me a card for a fee. Another university required me to donate to a non-profit called Friends of the Library or some such and then they gave me a card in appreciation for my donation. These were in Arizona and Utah respectively.
So, if one can access the OED with a university card and if one can find one of these schemes to give one a card then one should, perhaps, have OED access without fussing with the public library.
BTW, ask more than one person. The people working at the University of Utah library knew little or nothing about Friends of the Library. I learned of it via some alumni mailing which I go even though I was not an alumni.