Sully, a 4-year-old border terrier mix who looks like “an oversized Chihuahua with giant legs,” found the couple who would eventually become his new owner — but it wasn’t because they adopted him. First, they fostered him.

Tyler Gray said he and his wife, Sarah, were providing foster care to six dogs, including Sully, over Christmas. Somehow, the small dog managed to find a way into their hearts and the couple decided to keep him. Ironically, in the rescue community, it’s called a “foster failure,” when people decide they can’t part with the pet they’ve taken in temporarily and instead decide to adopt it.

In the past decade, providing cats and dogs, particularly those who are scheduled to be euthanized at kill shelters, with a temporary home until a family adopts them has become popular in the Baton Rouge area.

Companion Animal Alliance, which runs the East Baton Rouge Animal Control and Rescue Center, has more than 75 foster families willing to help deal with the overwhelming influx of animals the center receives. In just the first quarter of 2015, the city-parish shelter took in 859 dogs and 329 cats.

Smaller, nonprofit rescue organizations such as Yelp! Baton Rouge, Capital Area Animal Welfare Society and Friends of the Animals have started foster programs to help get dogs and cats out of the city-parish shelter and into homes.

CAAWS and the city-parish shelter offer the option of foster care for cats and dogs, while Yelp! and FOTA’s programs focus only on dogs.

And they’ve begun to make a dent. In the first quarter of 2015, rescue organizations took in 245 dogs and 86 cats from the city-parish shelter, according to CAA’s website.

“Nobody was fostering 10 years ago,” said Paula Schoen, president of Friends of the Animals.

When a person decides to foster a dog, the rescue organization pays all of the costs, including vaccinations, food, crates and anything else the animal might require. Only the city-parish shelter, due to what officials call budget limitations, asks foster families to pay for food and any toys they may wish to purchase.

If the dog needs training, the city-parish shelter provides it free through Fleur de Lead Dog Training.

The parish shelter had 500 dogs and cats that started out in foster care and ended up being adopted last year, said Lily Yap, the city-parish Animal Control Center foster and rescue coordinator. During July, the shelter found homes for about 40 dogs and cats through its foster program, she said.

“Being able to stay in homes makes animals more adoptable, allows for more kennel space at the shelter for incoming animals and simply increases their well-being until they find forever families,” Yap said.

Some animals have special requirements like a fenced-in yard or just need more attention, and the foster programs try to match families with the right animal.

Most foster families take one animal at a time, but there are a few like the Grays who will take in more.

The couple is what CAAWS Adoption Coordinator Sabra Smith calls “repeat fosters.”

“Once they get a dog adopted, they turn around and pick up another one and then take it home,” Smith said.

They have provided foster care to about 15 dogs in the past year, Gray said.

“We have some that take litters of puppies,” Schoen said. “We have some that have three dogs. It’s just whatever you feel comfortable with.”

If they do need extra care like being housebroken, a foster family can help, said Jessica Card, executive director of Yelp! Baton Rouge, a local no-kill rescue organization started in 2009.

“(A foster family is) helping that dog become more adoptable by teaching it rules and boundaries,” Card said.

Card said Yelp! has an average of 12 to 15 families fostering dogs at any given time, with 20 dogs in the foster program at a time. The rescue organization had 250 adoptions last year and has achieved 99 adoptions so far in 2015.

If a foster family already has pets, the rescue groups generally require proof of vaccinations and will set up a meeting for the foster animal and the family’s pet before the family officially takes on the animal.

FOTA recently began letting families take animals home over a weekend to see how they adjust to each other.

Unlike other organizations, FOTA does not have a shelter but depends on foster families. FOTA has an “adoption house” where the dogs are taken each day so they can be seen and possibly adopted. Afterward, the dogs go home with their foster families. Schoen said the adoption house holds 30 to 35 animals during the day and 20 to 30 people provide foster care on a regular basis. FOTA has about 45 dogs adopted each month that it gets from the East Baton Rouge shelter.

“Our weekend fostering has been huge for us, huge,” Schoen said.

CAAWS has about seven animals in foster homes at any time. The organization takes in animals from the parish’s shelter as well as two kill-shelters in Jefferson Parish. CAAWS’ shelter can hold 15 dogs. Though the organization started in 1979, Smith said foster care has become more popular at CAAWS.

“I started doing volunteer work with CAAWS about six years ago, and we’ve definitely seen an increase in people who want to foster,” Smith said.

Foster care allows rescue groups to learn more about the dogs and to pass that information along to a perspective owner.

“The dogs act differently in a kennel environment than they do in someone’s home,” Card said. “Some who are high energy at the Yelp! house and maybe seem like they don’t have manners, you get them in a home and they just relax. ”

Gray said Sully is dramatically different now than when he was first brought to their home.

“He is not scared of people anymore,” Gray said.

And to those who wonder if it’s upsetting for foster families to let go when their guests get adopted, Schoen said, “Yes and no.”

“A small part of you leaves with the dog, but most keep up with their fosters,” she said. “And I guarantee almost everybody has in the back of their mind who they can save next.”

Follow Danielle Maddox on Twitter, @Dani_Maddox4.