The Hillsborough County Pet Resource Center is the only open-admissions pet shelter in the county, meaning they take in pets regardless of breed or age. Because the shelter is open admissions, they're often overcrowded. They currently have about 300 dogs and 140 cats, which puts them well over capacity.
The Pet Resource Center is always looking for people to adopt but also people to foster pets for a couple days or weeks. They even have the Adventure Tails program, which allows people to take a dog or cat for the day and return it before the shelter closes that day. Even such a short period of time gives the pet a break from the shelter and the person taking the pet on a trip can give the shelter useful info like how the pet did in the car, did they react to other pets, etc.
You can find more information at HCFL.gov/pets.