I'm not aware of any libraries that offer non-resident library cards, but Tennessee actually did something right for a change and made one large ebook library for the entire state. Thanks to this, even if you live in a small town here, you have access to a large selection of ebooks, audiobooks, magazines, and videos through
R.E.A.D.S. (Regional eBook and Audiobook Download System), which is run by Overdrive. Looks like there's 77,581 Kindle titles, 72,550 epub & 22,164 audiobooks.
They even made it incredibly simple to use. Your local library card is your login, so I just select my library and type in my whopping four digit card number and have full access. I noticed they've added a mobile phone number verification as well, somehow using it to determine if your zip code's in the state.