Nancy, my guess is that you'll only have digital access to publications that the library of the district to which you belong (i.e., the district to which you pay taxes) subscribes.
Based on my address, I pay taxes to the Cook library district, which covers parts of Libertyville, Vernon Hills, and Mundelein. I have also registered my library card at the Fremont library in Mundelein and the Highland Park library, so I can check out print materials from there, but I am not able to access their online materials. The (predominantly Libertyville) Cook library subscribes to a set of online magazines and other digital resources that is somewhat different from what the Fremont (predominantly Mundelein) library subscribes to. Cooks Illustrated is not among the online magazine subscriptions I have access to.
So my guess is that you are limited to the set of digital subscriptions paid for by whatever library corresponds to the district that you pay your taxes to. If that library does not subscribe to Cooks Illustrated, I fear you are out of luck, unless you can badger the library's suggestion box and board members into adding Cooks Illustrated to their subscription package. But given how profit-focused (shall we say) Cooks Illustrated is, I wouldn't be surprised if it didn't make online access available to public libraries at a reasonable price.
May I suggest, Nancy, that you do what I did and switch from a paper Cooks Illustrated subscription to a digital subscription. The latter gives you access to the archive of all previous editions.
I hope this is helpful.
Edited to add: I mean digital subscription specifically to Cooks Illustrated only, not the all-access option, unless you know for sure that you also want digital access to Cooks Country and ATK in addition to CI.
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