After you have rescued your first pet, you will be able to access the Shelter section where, in addition to the animals that are already safe, those that still need your help will also be listed.
You will find this section of the game by pressing and holding on any of the pets. This will open two tabs: Shelter and Missing. In the Shelter tab, you will find the pets that are visible in the Shelter, while in the Missing tab, you will see the pets that you have not yet rescued. You will also be able to modify their names here.
We invite you to visit the article Can I change the names of the pets?