Last year, seven pets died during flights, according to Transportation Department data. Of these fatalities, six occurred on Hawaiian Airlines flights. Five of these animals were short-faced dogs shipped in cargo, said Marissa Villegas, a Hawaiian Airlines spokeswoman.

Ms. Villegas said that before traveling on Hawaiian, passengers traveling with pets are required to acknowledge and accept the risks.

Private carriers step in

Meanwhile, private jet companies have stepped in, offering air travel for pets inside the plane without having them spend hours trapped in crates or carriers. Some, like NetJets and VistaJet, have dedicated pet programs. On VistaJet flights, pets are fed prime cuts of meat, offered toys and given a soft mat to sleep on.

Leona Qi, president of VistaJet U.S., said there’s been a steady rise in the number of people traveling with their pets and that about half of the company’s customers bring their animals along. Most are dogs, but she said there have also been rabbits, falcons and once, a chameleon. VistaJet sells memberships that offer different levels of access to flights. Madelyn Reiter, a spokeswoman for the company, declined to detail prices.

Ms. Qi said that the pet program had frequently been “the determining factor” in convincing clients to upgrade to more expensive memberships which offer more cabin space.