Hundreds of thousands of travelers hoping to fly on Southwest Airlines this week are likely to be left in limbo a few days longer.

After canceling roughly 13,000 flights in the last few days, the airline is planning to remain on a reduced flying schedule for a few more days, its CEO said in a statement late Tuesday.

All domestic airlines have returned to pre-storm delay and cancellation levels after being knocked off-kilter late last week by a severe winter storm. Yet Southwest Airlines, plagued by staffing shortages and an outdated scheduling system, is still paralyzed.

As of 9:30 a.m. ET on Wednesday, Southwest had canceled 2,508 flights for the day, accounting for 62% of its scheduled flights, according to FlightAware.

Thursday isn't looking great either: 2,348 Southwest flights have already been taken off the departure boards.

Southwest CEO admits a system upgrade is overdue

Denver International Airport and Chicago Midway International are registering the highest number of cancellations.

But lines of passengers hoping for a rebook are snaking around Southwest ticket counters from coast to coast. Airports in Baltimore, Las Vegas, Nashville, Dallas, Phoenix and San Diego are all registering more than 80 departing flight cancellations from Southwest and other airlines.

In a video statement, the airline's CEO, Bob Jordan, said Southwest is aiming to "be back on track before next week."

"After days of trying to operate as much of our full schedule across the busy holiday weekend, we reached a decision point to significantly reduce our flying to catch up," he said.

"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."

Department of Transportation promises to hold Southwest accountable

Jordan also said he's been in touch with Transportation Secretary Pete Buttigieg about Southwest's customer service commitments.

Passengers say lack of communication from the company has been especially frustrating. Some didn't know their flight had been canceled until they happened to check the status online. Others say they waited hours to speak to a Southwest representative on the phone.

In an interview with PBS Newshour, Buttigieg described such practices as "unacceptable" and threatened to hold the airline to its own stated intentions.

"We're going to expect them to go beyond the letter of the law in terms of how they treat passengers, making sure they pay for things like hotels, ground travel expenses, meals and of course, refunds," Buttigieg said.

"I'm going to be watching very closely to make sure that they follow through."

It's not the first time the DOT has threatened to crack down on the air industry's operational shortfalls.

When thousands of flights were canceled due to staffing shortages in June, Buttigieg met with the CEOs of major domestic airlines and asked them to detail how they'd fix operations in writing.

Casey Murray, president of the Southwest Airlines Pilot Association, says the blame for Southwest's meltdown should be placed on the company's reliance on tech and procedures from the '90s, when the company was half its size.

"Employees were left to their own devices. [...] They weren't given the tools to do their jobs, nor were they given the leadership to answer questions and be able to provide solutions," he told NPR's Morning Edition.

The airline industry has long been wary of regulations, saying that they ultimately put the cost burden on the customer. And Murray agrees that Southwest is better suited than, say, the DOT, to decide what needs to change.

"But these passengers have to be taken care of," Murray said. "They have lost not only time and money, but memories with their families."

Under DOT rules, canceled flights must be promptly rebooked or refunded

Congress is also watching whether the airline will compensate customers. The Senate Commerce Committee will launch an investigation, its chair said on Tuesday.

Under current DOT rules, any passenger who saw their flight canceled is entitled to a prompt cash refund for the full cost, plus any extra purchases such as bag fees or a seat assignment.

Southwest says that all customers traveling through Jan. 2, 2023 have the option to rebook or travel standby without fees. But for those looking to get a cash payment, the company has opened a special self-service portal for processing.

Federal refund rules are still in a gray zone when it comes to significant flight delays or related costs such as hotel rooms or ground transportation. Southwest says it will fund these costs for overnight delays that were within their control, but it's unclear whether they'll honor that policy given the role weather initially played in the delays.

Democratic Sens. Edward Markey and Richard Blumenthal, who've proposed clarifying such rules during previous flight snafus, issued a statement on Tuesday calling for Southwest to issue "significant monetary compensation" to customers whose holidays "have been ruined."

"Southwest is planning to issue a $428 million dividend next year — the company can afford to do right by the consumers it has harmed," they wrote.