Go to your local newspaper, and to the newspaper of the nearest major city. Ask them if they can insert one of your flyers in every copy of their daily or Sunday paper. They may charge a small fee for the labor.
If your town, county, or state has a yearly fair in the fall, see about setting up a booth or station with some of the animals up for adoption. Rotate them out daily or twice daily. Make sure they have plenty of food and water. For dogs, walk them around the fairgrounds, making sure to clean up after them. Have your volunteers wear printed shirts with the name of the adoption agency in a large-size font, preferably printed on a very bright-colored T-shirt, to really stand out in the crowd.
If this is unfeasible, try a binder with information sheets on all the pets up for adoption. If anybody is interested, give them the address and directions of the facility itself.