If none of the strays catch your fancy or want a healthier pet, you can call the Sims Animal Shelter from your phone's 'Hire a Service' option. After installation of the Cats and Dogs Expansion Pack, strays will slowly be adopted among the NPC families, even those that are considered 'Out of World.' You may even want to risk a human Sim's needs and health if you're desperate for a specific stray, as strays tend to get adopted within a few days.
You will need to act quickly if you see a stray pet you really want.
