To unlock more animals, use puzzle pieces gained through the derby, mystery boxes and selected boats with the puzzle icon. There is a total of 6 animals: 2 giraffes, 2 hippos and 2 elephants.
You can see the progress and total of puzzle pieces collected in your Sanctuary Book.
Once you get an animal of the same kind, for example, a second elephant, you simply need to upgrade the animal shelter and Voila! The second animal will arrive to your sanctuary.