By Collin Eaton and Konrad Putzier

Employees of Phillips 66 got the message in mid-May.

The Houston-based oil refiner couldn't have a viable business if people didn't return to their daily commutes, Chief Executive Greg Garland said in an internal video seen by The Wall Street Journal. Phillips 66 would have to do its part.

"If we in the energy business are reluctant to go back to the office, why should other people go back? How can we get our economy going again?" he said in the video.

By early June, Phillips 66 had recalled the majority of its employees to its main Houston office, which houses around 2,300 people. It has stayed mostly full as Texas has emerged as a hot spot for new infections that have strained the state's hospital system. Employees at the company have been discussing what they describe as lax safety protocols on issues like masks and an onerous approval process to work from home, according to several people familiar with the conversations.

Phillips 66 spokesman Dennis Nuss said the company has continuously updated protocols based on direction from the Centers for Disease Control and Prevention, state and local guidance and employee feedback, and he didn't dispute Mr. Garland's video message. The company has said its federal designation as critical to U.S. infrastructure exempts it from state rules on workplace capacity. Phillips 66 "understands that some employees have concerns" and made accommodations for high-risk employees and caregivers of high-risk family members to work from home, Mr. Nuss said.

"We firmly believe that we have a safe workplace which allows us to continue with an essential business for the nation," he said. "The practices put in place at Phillips 66 demonstrate that it is possible to both reopen many of our businesses while protecting the health and safety of the workforce."

As parts of the U.S. economy attempt to reopen, companies and institutions are taking various approaches to getting people back to work as coronavirus infections surge across the U.S. With more than 3.3 million confirmed cases nationwide and a death toll topping 135,000, employers say they are operating under unprecedented circumstances with no universal guidelines, making it difficult to balance workers' safety and financial exigencies after months of inactivity. The result is a segment of the workforce ranging from baseball players to bankers who say they feel undue pressure to report for duty.

Employees of Crédit Agricole Group received an email on June 8 from Marc-Andre Poirier, senior regional officer for the Americas at the financial firm's corporate and investment-banking unit, saying that all New York City employees would start returning to the office in phases beginning on June 15. Some would be told to return full-time while others would rotate in and out of the office in assigned groups, according to the email, which was viewed by the Journal.

Employees with pre-existing medical conditions were told to contact human resources. One employee who went through the bank's process of asking not to return to the office due to an elevated health risk was told the bank saw no reason for him not to return, according to an email viewed by the Journal. If he decided his health risk prevented him from returning, he was told, he could use vacation days, request unpaid time off or apply for short-term disability leave.

"There's pressure for people to go in," said another employee. Aside from health risks, the return process created logistical challenges. Employees who were supposed to return on June 15 were notified days before, leaving parents with little time to arrange for child care.

A document titled "Frequently Asked Questions" attached to Mr. Poirier's June 8 email said that with respect to child care, "It is each employee's responsibility to make sure he or she is able to come back to work when required to do so."

A Crédit Agricole spokeswoman said the firm has made every effort to protect its employees during the pandemic. She said it is closely regulating the number of employees in the office and conducting health screenings by using questionnaires and on-site nurses. She added the firm doesn't comment on the private health of its employees but that it is complying with all laws related to employee health, privacy and safety.

Since Phillips 66 began calling workers back to work, Covid-19 cases in its home state have surged. Texas has reported a total 275,058 positive cases as of Tuesday, with about 77% of those emerging since the end of May, according to the state's health department. The state also hit a record in new daily cases Tuesday, with a single-day increase of 10,745, the data show.

Several other large energy companies in Houston, including Chevron Corp., BP PLC, Royal Dutch Shell PLC's U.S. subsidiary Shell Oil Co. and Total SA, have kept most of their employees working remotely throughout the pandemic, with some pulling those who had returned to offices back home as the fourth-largest U.S. city became a coronavirus hot spot.

Phillips 66's Mr. Garland said in the May video that employees had proved they could work remotely, but staying home until a vaccine was available wasn't an option for the economy, the energy industry or the company. The effects of the pandemic could cost the company $3.2 billion in the first half of the year, he said. In addition, he said limiting workers to teleconferences and video meetings would make it impossible to tap into the creativity and productivity generated by office encounters.

The company has had 39 cases of Covid-19 at its Houston headquarters so far, representing less than 2% of the workforce there, according to an internal dashboard viewed by the Journal.

Mr. Nuss, the spokesman, said Phillips 66 has monitored, isolated or quarantined employees as necessary. As of July 10, it hadn't seen any evidence of workplace spread "when employees adhere to our protection practices," and contact tracing shows most positive cases result from exposures outside the headquarters office. None of the known cases have resulted in hospitalizations, he said.

When its office reopened, Phillips 66 didn't require employees to wear masks in several common areas until late June, employees said. In mid-June, when Harris County ordered businesses to mandate face coverings, the company said in an email viewed by the Journal that the order applied to businesses that provided goods and services directly to the public, which the company doesn't do at its corporate headquarters.

When the county raised its coronavirus threat level to the most severe near the end of the month, the company told employees in an email it wouldn't be changing its office staffing levels. It adjusted its policy on masks to require all employees and contractors to wear face coverings in common areas, when people weren't sitting at their desks or in meetings where social distancing was inadequate.

Employees said some managers have encouraged workers not to criticize leadership decisions.

"We are unaware of any such instructions or directives by individual managers," Mr. Nuss said. The company also "rejects any suggestion that Phillips 66 discourages the input of its employees, positive or negative, " he said.

Since March, Mr. Nuss said, the company's emergency operations center has routinely evaluated and adjusted its plans and protocols based on employee feedback and authorities' guidance. The company has surveyed employees, who get weekly updates from executives through internal videos, podcasts and virtual meetings. It also has a social-distancing assessment tool, a mobile app for health screening and an email address for employee feedback.

Employees who live with or are themselves medically vulnerable can seek the company's high-risk designation to work remotely, after submitting documentation from their health-care providers, in line with the Americans With Disabilities Act, Mr. Nuss said. Many employees who have requested that status were identified and have worked from home, he said.

In internal communication to employees on the company's pandemic pay policy, Phillips 66 said primary caregivers of children attending closed schools or day care would be able to work remotely for an additional workweek to find alternate child care following a business unit's transition back to the office. Employees unable to work remotely could take up to one week of paid time off to find alternate child care.

Mr. Nuss said company officials are reviewing the policy and "are mindful of the upcoming school year next month." Texas education officials said last week they plan for schools to open, with opportunities for some virtual instruction and the expectation that closures could become necessary.

--Patrick Thomas contributed to this article.

Write to Collin Eaton at collin.eaton@wsj.com and Konrad Putzier at konrad.putzier@wsj.com