Southwest Airlines Co. showed progress Saturday in its push to regain credibility with regulators and travelers, especially those whose holidays were disrupted by the company’s meltdown over the past week, but cancellations increased late in the day.

The Dallas-based carrier had 30 Saturday flight cancellations as of Saturday evening, according to FlightAware. Overall, more than 250 flights among all airlines flying to, from or within the U.S. had been canceled. Southwest’s total compared with 15 for United Airlines and 11 for Delta Air Lines.

A Southwest spokeswoman said earlier in the day that the airline was operating a normal Saturday schedule of about 3,400 flights. Meanwhile, the carrier was seeking volunteers among its employees to help the customer-service staff catch up with requests for refunds and reunite customers with missing bags.

In a video distributed to staff members Friday, Southwest executives were upbeat about the near-term outlook. “I’m just very pleased to share that things are going very, very well,” said Bob Jordan, the airline’s chief executive. 

Andrew Watterson, chief operating officer, said that lines had grown shorter and that the airline expected to provide normal service during the New Year holiday period and beyond. In another update Saturday, he said Southwest had deployed “an army” of people to ship bags back to customers, in some cases using UPS and FedEx to transport lost luggage. 

Southwest has ramped up its service after a meltdown that resulted in nearly 16,000 canceled flights between Dec. 22 and Dec. 29. Those cancellations, stemming from the recent winter storm, left thousands of holiday travelers stranded, furious and in many cases separated by hundreds of miles from their luggage.

Though the storm created problems for all airlines, Southwest canceled far more flights and was much slower than others to recover. Executives of the airline have said the scheduling system used to revise crew schedules after storms was overwhelmed by the volume of changes required. Airline staff members fumbled with makeshift manual methods to match up available crew and planes.

Southwest Airlines travelers waited for luggage in Minneapolis on Friday.

Photo: Abbie Parr/Associated Press

To get back on track, the airline shrank itself for much of this week, operating roughly a third of its typical schedule on Tuesday, Wednesday and Thursday as it worked to get crews and planes back in place. The airline resumed operating its full schedule Friday. 

Southwest’s problems are far from over. Regulators, lawmakers and union leaders have said they are monitoring the airline’s response to the crisis. Southwest has apologized repeatedly and promised to reimburse affected travelers.

“As SWA turns the corner operationally, focus must remain on promptly compensating passengers caught in last week’s breakdown,” Transportation Secretary Pete Buttigieg said in a tweet Saturday.

One regular Southwest customer who still needs more reassurance is Allison Whitney, a professor of film and media studies at Texas Tech University. She was due to fly home to Lubbock, Texas, from Minnesota on Wednesday, but her Southwest flight was canceled. Facing the risk of being stranded until early in the new year, she booked an American Airlines flight Friday and made it home. 

Ms. Whitney likes Southwest’s luggage and easy-rebooking policies and finds that it can be the only good choice for some of her trips. But she said that after this week, she might hesitate to rely on Southwest for longer trips until she is convinced that the airline’s computer systems are up-to-date.

Write to James R. Hagerty at bob.hagerty@wsj.com and Alison Sider at alison.sider@wsj.com