The reason that only airplane mode works for Kindles is that
Amazon devices are set up with a sort of cell-phone-type calling that only works to communicate with one place: Amazon. This means that Amazon doesn't need wifi to reach the device. Overdrive access to Kindle devices is routed through Amazon. However, as PP have said, airplane mode will preserve the loan for the extra time.
Please be aware that most library systems have multiple borrowing period choices, but the default setting is usually the shortest loan period choice. Check your Overdrive library account settings to be sure that you are borrowing for the longest possible period if you are not a very fast reader. (And if you get a few chapters in and find you don't like the title, please "return" it right away, rather than letting the loan expire. Someone else may be waiting for the book.)
I'm a librarian and I personally have two Overdrive accounts plus one on HooplaDigital. I average about 30 electronic loans a month, and it's not unusual for me to run through 2 fiction titles in a day (I also borrow audiobooks.) What we typically tell people to do is to reserve a variety of titles on a staggered basis, so as to keep a regular flow of books that interest you. You can always "return" what you decide you don't like.
(If this seems counter-intuitive, realize that funding for particular Library programs typically is demand-driven. If you want to provide a larger collection, you have to justify that by proving that the demand is high enough to warrant putting a larger share of money into that program cost. We can't prove that unless we get traffic statistics that show it.)