I personally have found two simple methods for getting into libraries. There may be many other methods, but this is what has worked for me.
#1 Contact the library directly with a letter or email. I have even extended this with my personal sales brochure that lists all of my books. These emails, letters, and brochures have even resulted in speaking engagements with different libraries.
#2 Have your friends and family in other areas go to their local libraries and request your book. It’s that simple. In most library systems, they don’t even have to physically go to the library to make this request. They can do so online as long as they have a library account.