(CNN) — Relief is still a few days away for passengers booked with Southwest Airlines this week, as the beleaguered airline continues to grapple with what US Transportation Secretary Pete Buttigieg has referred to as a complete meltdown of the system.

Out of the 2,790 cancellations already made for Wednesday flights within, into, or out of the United States as of 10:30 a.m. ET, 2,508 of them are operated by Southwest, according to flight tracking website FlightAware.

In all, Southwest has canceled more than 15,700 flights since winter weather began disrupting air travel on December 22. That figure includes more than 2,300 flights already canceled for Thursday. Other US airlines have since recovered from the storm disruptions.

Denver International Airport is leading the way Wednesday in the number of cancellations. At Chicago Midway and Dallas Love Field, close to half of Wednesday flights are canceled.

Southwest plans to fly a reduced schedule over the next few days to reposition crew and planes, airline CEO Bob Jordan said in a video released by the airline late Tuesday. "We're optimistic to be back on track before next week," Jordan said.

Buttigieg says he spoke directly to Jordan on Tuesday about the thousands of flights that have been canceled this week.

"Their system really has completely melted down," Buttigieg told CNN's Wolf Blitzer on Tuesday.

"I made clear that our department will be holding them accountable for their responsibilities to customers, both to get them through this situation and to make sure that this can't happen again."

Those responsibilities include providing meal vouchers and hotel accommodations for passengers whose flights were disrupted "as a result of Southwest's decisions and actions," a Department of Transportation spokesperson said Tuesday.

US airlines are also required to provide cash refunds to passengers whose flights were canceled and opted not to travel, the DOT said.

Tuesday at a glance

More than 3,200 flights within, into or out of the United States were canceled on Tuesday, according to FlightAware.

Of those canceled flights, some 2,694 were those of Southwest — a stunning 84 percent of all canceled flights in the United States.

Long lines of travelers attempting to rebook or make connections were witnessed at Southwest ticket counters at multiple US airports on Tuesday, while huge piles of unclaimed bags continued to grow as passengers struggled to reclaim their luggage in airports including Chicago's Midway International, Harry Reid in Las Vegas, and William P. Hobby Airport in Houston.

Southwest Airlines does not have interline agreements with other carriers that would allow its agents to rebook passengers on a different airline, narrowing the options for stranded passengers.

"Southwest is unique in the industry in that we don't have codeshare partners. That is just part of our business model," a Southwest spokesperson told CNN.

Passenger Trisha Jones told CNN at the airport in Atlanta that she and her partner had been traveling for five days, trying to get home to Wichita, Kansas, after disembarking from a cruise at Fort Lauderdale, Florida.

After her flight out was canceled, she stayed with relatives, then rerouted to Atlanta to pick up a connecting flight.

"We were fortunate, because we were in Fort Lauderdale — my family lives in the Tampa bay area so we were able to rent a car to go see my family for Christmas," Jones said. "We've seen a lot of families who are sleeping on the floor, and it just breaks my heart."

Buttigieg: 'A lot of cleaning up to do'

Southwest has blamed the travel disaster on several factors, including winter storm delays, aggressive flight scheduling, and outdated infrastructure.

"From what I can tell, Southwest is unable to locate even where their own crews are, let alone their own passengers, let alone baggage," said Buttigieg, adding that he also spoke with leaders of the airline's unions representing flight attendants and pilots.

The secretary said he told CEO Jordan that he expects Southwest to proactively offer refunds and expense reimbursement to affected passengers without them having to ask.

"I conveyed to the CEO our expectation that they are going to go above and beyond to take care of passengers and to address this," he said.

Buttigieg told CNN the Department of Transportation is prepared to pursue fines against Southwest if there is evidence that the company has failed to meet its legal obligations, but he added that the department will be taking a closer look at consistent customer service problems at the airline.

"While all of the other parts of the aviation system have been moving toward recovery and getting better each day, it's actually been moving the opposite direction with this airline," said Buttigieg.

"You've got a company here that's got a lot of cleaning up to do," he said.

Southwest CEO issues video apology

Jordan apologized to passengers and employees in the video released on Tuesday evening.

"We're doing everything we can to return to a normal operation, and please also hear that I am truly sorry," Jordan said.

He said with large numbers of airplanes and flight crews "out of position" in dozens of cities, the airline decided to "significantly reduce our flying to catch up."

While Jordan acknowledged problems with the company response, the statement suggested that he did not foresee massive changes to Southwest's operating plans in response to the mass cancellations.

"The tools we use to recover from disruption serve us well 99 percent of the time, but clearly we need to double-down on our already-existing plans to upgrade systems for these extreme circumstances so that we never again face what's happening right now," said Jordan.

Is there anything passengers can do?

Southwest has warned that this week's cancellations and delays are expected to continue for several more days.

So what should customers do?

"First things first, travelers who are still stuck waiting on Southwest and need to get somewhere should try to book a flight with another airline as soon as possible ... right now, really," said Kyle Potter, executive editor at the travel advice website Thrifty Traveler, in an email to CNN Travel late Tuesday afternoon.

"Every airline in the country is jam-packed right now, so your odds of even finding a seat — let alone at an even halfway decent price — get smaller by the hour," Potter said.

"Travelers in the thick of this should be sure to save all their receipts: other flights, a rental car, nights at the hotel, meals, anything," Potter said.

If you've been left in the lurch and your efforts to reach a customer service agent are going nowhere, the founder of Scott's Cheap Flights suggests trying an international number.

"The main hotline for US airlines will be clogged with other passengers getting rebooked. To get through to an agent quickly, call any one of the airline's dozens of international offices," Scott Keyes said. Those agents can handle US-based reservations, Keyes said.

Click here to get international numbers that Southwest has previously posted.

Southwest: 'Keep your receipts'

Southwest spokesperson Jay McVay said in a news conference at Houston's William P. Hobby Airport on Monday night the airline will do everything possible to right the challenges passengers have experienced, including "hotels, ride assistance, vans ... rental cars to try and make sure these folks get home as quickly as possible."

He promised that all customers, even those who had already left the airport or made alternate arrangements on their own, would also be taken care of.

"If you've already left, take care of yourself, do what you need to do for your family, keep your receipts," McVay relayed. "We will make sure they are taken care of, that is not a question."

What's wrong from a pilot's point of view

Speaking to CNN on Tuesday, the vice president of the Southwest Airlines Pilots Association, Capt. Mike Santoro, said the problems facing Southwest were the worst disruptions he'd experienced in 16 years at the airline.

He described last week's storm as a catalyst that helped trigger major technical issues.

"What went wrong is that our IT infrastructure for scheduling software is vastly outdated," he said. "It can't handle the number of pilots, flight attendants that we have in the system, with our complex route network.

"We don't have the normal hub the other major airlines do. We fly a point-to-point network, which can put our crews in the wrong places, without airplanes."

He added: "It is frustrating for the pilots, the flight attendants and especially our passengers. We are tired of apologizing for Southwest, the pilots in the airline, our hearts go out to all of the passengers, they really do."

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