Last November, Howard Schultz traveled to Buffalo, N.Y., with a message to workers considering forming a union: Starbucks would do the best job looking out for employees’ interests.

During an hourlong speech delivered without notes, Mr. Schultz, Starbucks Corp.’s former leader, appealed to employees to work together with management. “Our core purpose and reason for being is to build the kind of company that creates a fragile balance between profit and doing the right thing,” he said.

Many baristas in Buffalo didn’t stand with Mr. Schultz; they voted to unionize instead. They have said they believe organizing is needed to push for better pay, adequate staffing and a louder voice in company affairs.

Now Mr. Schultz, 68 years old, will need to please shareholders while also trying to win over Starbucks employees—and those the chain still aims to hire. In April, Mr. Schultz will return for his third stint running Starbucks, taking on an interim chief executive role after the chain this week said current CEO Kevin Johnson would retire after five years atop the company. Part of Mr. Schultz’s focus, he told employees this week, is establishing a new tone with the roughly 230,000 workers staffing its U.S. cafes.

“We have to take a hard look at how we are doing as a company, and as a community of partners,” Mr. Schultz wrote in a message to Starbucks workers Wednesday. “These are the questions on my mind as we renew a journey to see beyond what Starbucks has been and what it is today to what it will become.”

Starbucks’s Howard Schultz: From Coffee Entrepreneur to Three-Time CEO

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Across the U.S., restaurant operators are struggling to recruit and retain enough staff to fully staff locations, and quit rates among hospitality workers hit record highs last year, federal data showed. Starbucks, the world’s biggest coffee chain in terms of sales, had to reduce hours and limit services earlier this year at some locations that were short-handed because of continuing worker shortages and rising Covid-19 cases.

Thinly staffed stores have left some baristas feeling overwhelmed, workers have said, while customers continue to order highly customized, labor-intensive drinks.

Starbucks needs more workers to meet its growth goals. The company this week said that it aims to open more than 20,000 cafes globally by the end of the decade. It also needs to improve investor sentiment toward the stock. Starbucks stock, which has trailed other restaurant company shares in recent months, climbed roughly 8% since Mr. Schultz’s return was announced Wednesday.

After buying out the local Starbucks coffee outpost in 1987, Mr. Schultz molded Starbucks’s image and customer experience, developing its cafes as a place to linger over espressos and lattes. He also helped design the company’s offerings for its baristas and other workers, offering benefits that Mr. Schultz said went beyond most retailers—covering the cost of employee healthcare, time off and benefits for those working at least 20 hours a week at the chain. A year before the company went public in 1992, Starbucks started giving baristas stock, a practice the company has continued.

“I was convinced that under my leadership, employees would come to realize that I would listen to their concerns,” Mr. Schultz wrote in “Pour Your Heart into It,” his book about building Starbucks into a global coffee company. “If they had faith in me and my motives, they wouldn’t need a union.”

During his earlier stretches leading Starbucks, Mr. Schultz often held town hall-style forums to speak with workers and sometimes directly reached out to employees. In 2016, after a California barista named Jaime Prater wrote an online petition about working conditions at Starbucks that drew nearly 25,000 signatures, Mr. Schultz called the employee. The CEO later bumped up pay, and company stock issued to many U.S. cafe workers.

Mr. Prater recalled feeling thrilled when Mr. Schultz called him, feeling like his voice mattered. He said he appreciated the additional benefits, but said the overall working environment at his cafe didn’t improve.

“It was the most demanding and hardest job I’ve ever worked,” said Mr. Prater, who left Starbucks in 2018 and now works as a marketing manager.

The company is now dealing with the most concerted unionization campaign in its history. Pro-union workers have said unionizing will help deliver better pay and working conditions, and representation among Starbucks’s decisionmakers. Starbucks has said it respects the unionization process, but wants to maintain its direct relationship with workers and has advised them against organizing.

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Since the union drive began at Buffalo-area cafes last year, around 140 U.S. Starbucks cafes have petitioned to hold votes on whether their workers should be represented by Starbucks Workers United, formed in the run-up to the Buffalo cafes’ votes in late 2021. Next week, the National Labor Relations Board is slated to count ballots in the first union drive in Mr. Schultz’s primary home of Seattle. Starbucks has petitioned for the federal labor agency to review that vote’s structure, as it has in other elections. So far the company’s appeals haven’t succeeded.

Mr. Schultz and the company’s next permanent CEO, whom Mr. Schultz will help select, must respond to the union push and workers’ concerns in credible ways, said Starbucks board Chairwoman Mellody Hobson. The company also needs to hold itself accountable when it gets something wrong, Ms. Hobson said in an interview.

“I feel very strongly that our partners were speaking to us, and I don’t think we were hearing that,” said Ms. Hobson, referring to the company’s baristas.

Taylor Shaw, a shift supervisor at a Buffalo Starbucks that recently voted to unionize, said she hopes Mr. Schultz’s return will lead to a greater focus on workers.

“It’s been, ‘What are your drive [through] times? What are your sales?’ Obviously that’s important when you’re running a business, but it just seems like Howard was more concerned about the personal connection that Starbucks talks about,” Ms. Shaw said.

Mr. Schultz is inviting worker feedback. He signed his Wednesday email to employees asking them to email him at his Starbucks address.

—Stephen Council contributed to this article.

Write to Heather Haddon at heather.haddon@wsj.com