Steve Frost, a retired fire captain in Northern California, said he saw the photo and thought Elmo “looked like hell.” He immediately decided he wanted the dog.

Four months later, Frost sits by his fireplace every morning and evening and gives Elmo four pills for his various ailments, “like an old man.” On Wednesday morning, he took Elmo in for prostate surgery. Frost, who had not owned a dog in several years, is now ushering one through its final years of life, which he says he figures will be “a lot better than living in a kennel.”

Frost, 59, met Elmo through the Thulani Program, one of a growing number of animal organizations focusing on adopting out older dogs, or “senior dogs” that are typically 7 years or older. Their age makes them some of the hardest-to-place animals in a society that still adores romping puppies, although that is changing as books on elderly dogs and social media campaigns convince pet-seekers that the mature pooches often come with benefits, such as being house-trained, more sedate and less demanding of people with busy lifestyles.

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But some of those adopters go further, selecting pets from programs for dogs in need of hospice care, or what amounts to assisted living for very ill or very old dogs. These programs usually commit to covering the cost of a dog’s medical and dental care, which might otherwise be a major obstacle to finding them homes, said Lisa Lunghofer, executive director of the Maryland-based Grey Muzzle Organization. The donor-funded group gave $225,000 in grants last year to 38 senior dog programs nationwide, several of which now promote hospice adoptions.

Frost, who lives in Redding, Calif., and is a part-time professional pilot, said he knew he wasn’t up to the task of raising a puppy. He also knew he wanted a German Shepherd. An Internet search led him to Thulani, and that led him to Elmo, one of the organization’s hospice dogs.

Frost knows little about Elmo’s past, other than that he was turned over to an animal shelter in Los Angeles and had clearly been neglected. His ears had mites, his innards had worms, his prostate had a tumor and he was puppy-like in one key way: At age 11, he wasn’t house-trained. Now Elmo has two beds in Frost’s home and a permanent place in the back seat of his four-door Ford F-150, and the two take what Frost called “a man shower” together every few days.

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“This guy has just burrowed his way into my heart,” Frost said.

Lunghofer stresses that most senior dogs do not require the kind of care Elmo has. The majority “just need a good home,” she said, and many “regain their vitality and reward their families with years of unconditional love and devotion.”

And more of them are finding those homes, she said. Grey Muzzle — which says it envisions “a world where no old dog dies alone and afraid” — recently surveyed its grantees, the majority of which said the situation for older dogs has improved in the past two years and that young people are more open to such adoptions. Nearly all said the main reason people adopt aging dogs is “altruism,” although mellowness and potty skills were also cited.

“This is a great way to ease into dog ownership,” said Erick Smith of Muttville Senior Dog Rescue in San Francisco, a Grey Muzzle grantee. “It’s not this epic commitment that you’re staring down.”

David Writz, 34, said he’s hoping his newly adopted 10-year-old black Lab mix, Dante — who is not a hospice case — will stick around for five years or so. Like Frost, Writz found his dog online after deciding he didn’t have time for a puppy. When the two met in person at Bob’s House for Dogs in Eleva, Wis., Writz was smitten, despite the fact that Dante was about 20 pounds overweight. Then the shelter called and told Writz that the dog would be having emergency surgery to remove an eye with glaucoma. Did Writz still want him?

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“I was like, ’Obviously,’ ” said Writz, who works in a payroll office. “I figured at the very least I’d just get him an eye patch.”

Knowing that Dante won’t be around for long “is the depressing aspect of it,” said Writz, who regularly takes Dante to a local brewery, where the dog happily begs for pretzels. “But I figure he’ll be happy the rest of his remaining years.”

Russell Ulrey, a Muttville volunteer who helped start the shelter’s hospice program, said he was initially worried that he wouldn’t find takers. He was wrong. Last year, Muttville adopted out 85 hospice dogs, and Ulrey said demand is higher than supply.

Caring for a terminally ill dog is “a life-changing experience,” said Ulrey, 41, but one he tells potential adopters to approach with flexibility. Ulrey, who has adopted several hospice dogs, said one lived 14 months until, one day, he charged up a hill to a favorite park and collapsed. A veterinarian euthanized him there. An Airedale mix named Ralph, in contrast, had multiple organ failure, rotten teeth and survived just a week.

“A dog like Ralph, we didn’t take him out to the park. He didn’t want to go,” he said. “We made him cozy and made him feel really loved, maybe for the first time in his life.”

These days, Ulrey and his partner, Marie Macaspac, are the parents of Chachito, a 20-pound mutt who qualified as a hospice dog because he is 16 years old, blind and deaf — exactly the kind of pet few adopters would be willing to take on. Chachito’s regime involves homemade meals of brown rice pasta and chicken, supplements for his joints, arthritis medication and lots of hanging out at the couple’s Fairfax, Calif., home.

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“He has his route,” Ulrey said. “He bonks into one wall and then knows he’s going to turn right.”

Farther north, in Redding, Elmo is recovering from his prostate surgery, which added hundreds of dollars to the $1,000 or more Frost has paid — with Thulani funds — to get the dog in shape. Soon, Frost said, he’ll take Elmo for a ride in an airplane.

“The best you can do is make him have a great life, because his life up until this point has been hell,” Frost said. As for the end? Frost said he doesn’t focus on that. “If it didn’t hurt, you’ve got to question the love that’s involved.”