The delays show that delivery networks such as UPS, which are critical components of the e-commerce boom, are still struggling to cope with the busiest shopping periods despite heavy investment to build out and automate their operations and capacity. For the first time this year, UPS tried to manage the flow of packages by adding surcharges to deliveries during peak periods.

The week after Thanksgiving is the start of the delivery crunch, as carriers begin to process online orders placed during the holiday weekend and millions more packages pour in, starting the following Monday. Adobe estimated that Cyber Monday sales hit $6.6 billion this year, up more than 17% from 2016.

ShipMatrix, a software provider that analyzes shipping data, said 89.2% of parcels shipped last week through UPS Express were delivered on the day they were promised. The worst on-time service hit packages that UPS picked up on Monday and Tuesday of last week.

By comparison, 99.4% of FedEx Express packages were delivered by the end of the day they were scheduled to arrive.

Mr. Gaut said UPS doesn’t comment on the accuracy of third-party data. UPS expects the “vast majority” of the 750 million packages that it ships between Thanksgiving and Christmas will be delivered on time, he added.

The U.S. Postal Service, which delivers more online orders to residences than anyone, is expanding delivery hours to handle the peak season’s volume, with packages delivered early in the morning, evening and on Sundays.FedEx Corp. is “well-positioned” to meet record demand during the peak season, when it plans to deliver up to 400 million packages, said spokesman Glen Brandow. “We plan and collaborate closely with our customers year round and engineer our networks to be ready to meet the significant surge in demand for residential deliveries,” Mr. Brandow said.

“We have planned for this holiday season all year long and have flexed out network and expanded delivery hours to accommodate increased volume,” said spokesman David Partenheimer.

UPS, based in Atlanta, is making some adjustments to cope. It has notified workers at more than 100 package-delivery centers where demand is highest that it may raise the number of hours drivers work to 70 hours over an eight-day period, up from 60 over seven days.

Union leadership, which represents UPS drivers, has objected to the move in some places, saying that the longer hours put package-delivery drivers at risk and keep them away from their families during the holidays.

In a letter sent Monday to UPS Chief Executive David Abney, Teamsters President James P. Hoffa said: “I fail to understand how the Company neglected to take the steps necessary to ensure that it had a sufficient number of trained workers available to meet the demand generated by the digital economy.” The Teamsters represent more than 250,000 UPS workers and are currently engaged in contract talks. 

UPS said the change complies with federal requirements and that workers are paid time-and-a-half for over eight hours a day.

“UPS customers can be confident that UPS is taking the necessary steps to ensure the network operates with its customary dependable performance throughout the holiday season,” Mr. Gaut said.

UPS has spent billions of dollars to add more warehouse space and automate sortation centers, where packages are sorted before being sent to shippers to reduce shipping time. It also has worked more closely with large retailers to avoid the problems of previous holiday seasons. In 2013, for instance, UPS was swamped by an unexpected glut of last minute packages in the days before Christmas and many shoppers didn’t receive their orders on time.

This year, UPS added three Boeing 747-8s, the largest commercially available freight aircrafts to its fleet, to help during the peak season. It also said it would hire 95,000 seasonal workers and even let some of them drive their own cars to deliver packages.

In October, Mr. Abney, the UPS CEO, told investors the company was “well-positioned to have a successful peak season, both from a customer standpoint and an investor standpoint. We’ve been working a long time on this, and I think there’s a lot of confidence throughout the group.”