Need to call an airline? Your hold time will be approximately one zillion hours

That simple chore, which he thought would take a few minutes by phone, turned into a three-day undertaking. A comic book illustrator, Mr. Maleev, 49, used the time to produce what he calls the Lufthansa Files. He sketched the Marvel Comics superhero Daredevil on the first day during his hourlong wait before he got disconnected. The next day, he drew another Daredevil while on hold for an hour, then hung up. On the third day’s hourlong wait, he sketched Marv from the “Sin City" comic series, and posted the drawings on Twitter.

“I can do this all day, apparently so can they," he tweeted. “Let’s see who blinks first."

Mr. Maleev never managed to reach anyone during those calls, he said, and eventually rebooked the flight by calling Lufthansa’s local office in Bulgaria.

Lufthansa said the wait times are due to high travel demand, complex booking requests and continuously changing travel regulations. “We are doing our utmost to increase our capacities to meet the demand," said spokeswoman Christina Semmel.

Passengers have returned to travel more quickly than many airlines anticipated, triggering delays and canceled flights due in part to staff shortages—and many customers who call in hopes of resolving issues have faced excruciatingly long hold times.

Often, they’re calling with requests that can’t easily be handled online. Many airlines say passengers are calling more often and with more complex questions about travel credits, flight changes or restrictions at their destinations. At the same time, airlines encouraged employees to take buyouts or retire early last year when it was unclear when travel would rebound. Delta Air Lines Inc. said its call center staffing was down by 50% this year, though the company said the situation has been improving.

The result has been telephonic purgatory for travelers.

Cam’Ron Wells, a 22-year-old reality TV producer who needed help changing a flight from Los Angeles to Miami after the website kept crashing, used what he was told would be a 280-minute wait time with JetBlue Airways Corp. to get through his to-do list.

Mr. Wells began, then finished, an episode of WE tv reality series “Love After Lockup," keeping the volume on his AirPods low in case an agent picked up early. He did two full cycles of laundry. He cooked some waffles and bacon. He washed the dishes. He packed for an unrelated trip for the next day.

He also set a timer for an hour and 45 minutes and went to sleep. “When I woke up from my nap, I was still on hold," he said.

The entire call lasted just over four hours, but his issue was resolved quickly. “Customer service was great, it was just: Oh my gosh, a four-hour wait for something that’s going to take three minutes," Mr. Wells said.

JetBlue attributed the high call volume to great travel demand and questions about Covid-19 requirements, and added that the company is hiring 4,000 crew members across the operation this year.

Ruby Murton said that she spent nine hours on hold with Delta in July, listening to the same few elevator music-like songs playing repeatedly, mixed with the occasional sound of an airplane taking off.

Ms. Murton, a 20-year-old student at the University of Michigan, turned her experience into a TikTok video. She needed to change her flight after Delta rescheduled her family’s trip to Detroit from Orlando, Fla., putting her on a separate flight that left hours before everyone else’s. After getting an estimated wait time of seven to eight hours earlier in the day, she was grateful when she called at 9 p.m. and was quoted a 5 hour, 53 minute wait. But the call stretched on, and she battled to stay awake.

“At four hours I had some Fruity Pebbles. This was the best part of my night. So delicious, except I added too much milk," says a peppy automated voice, like those in phone menus, in the video. “By hour seven I was questioning my entire existence."

She finally got through at 5:59 a.m. The worst part was not knowing how long the wait would be, said Ms. Murton. “I didn’t want to walk away from the phone for 30 seconds—what if they pick up and I don’t respond, and they hang up?"

During 23-year-old Nienah Saijas’s nearly eight-hour wait in July to reschedule a Delta flight, she made lunch, did a post office run—linking her phone to her car’s Bluetooth during the drive—and played house and techno music to cover up the repetitive tunes. Ms. Saijas, a personal assistant, said that the experience has deepened her patience: “If I have to wait 45 minutes or something on the phone, that’s nothing now."

Both Delta customers said they eventually had their issues resolved.

A Delta spokesman said the company is working to address the wait times. The airline has called back hundreds of former workers to help deal with the summer surge, and is hiring and training 1,300 employees to take customer calls by September. Delta has also contracted with a travel agency to handle some of the calls and said it’s on track to get down to single-digit minute waits. Some customers are offered the option to get a call back from an agent.

Delta Chief Executive Ed Bastian said there’s one more option. “If somebody needs help, just send me a note. I’ll take care of it," he said during the company’s July earnings call. Delta said he has received messages but declined to specify how many.