New York firm Milbank LLP on Thursday kicked off the latest round of salary wars, announcing pay raises of between 4.4% and 5.9% for lawyers without a partnership stake in the firm. The raises shifted pay for newly graduated lawyers to $200,000, up from a $190,000 benchmark that Milbank set in the last widespread round of raises three years ago.

There were signs Friday the new industry standard is likely to rise even further, with New York-based Davis Polk & Wardwell LLP telling its lawyers their pay would climb to between $202,500 for Class of 2021 law-school graduates and $365,000 for those who had been at the firm at least eight years.

Large law firms fared well financially during the pandemic. Many reported double-digit profit gains for partners, thanks to a boom in transactional work and drops in expenses as lawyers no longer flew around the world to meet with clients.

“It was almost unseemly to not pass some of that along” to the more junior lawyers, said

James Leipold,

the executive director of the National Association for Law Placement, which tracks industry hiring and pay trends.

Lawyers in the junior ranks often bill around 2,000 hours a year, which can require putting in far more than 40 hours a week when taking into account time spent on tasks that aren’t directly billed to clients. During the pandemic, many young lawyers worked in isolation in their homes, missing out on the social gatherings and late-night dinners at the office that once helped take the sting out of the workload.

“It’s been tough,” said

Scott Edelman,

the chairman of Milbank. “There’s no division between work and nonwork, and the recharging from social interactions.”

Mr. Edelman said the firm’s decision to raise pay followed a successful year for Milbank that saw revenue rise 15.6% to $1.23 billion and average partner profits increase nearly 16% to close to $4.5 million. He said an influx of work and rising cost of living drove the move.

Other firms that have made salary increases in the past two days include New York firm Cadwalader, Wickersham & Taft LLP, Boston-based Mintz, Levin, Cohn, Ferris, Glovsky and Popeo PC and 725-lawyer Proskauer Rose LLP. Many law firms have also paid associates bonuses during the pandemic.

Not every large firm will immediately race to match the new salary scale. For instance, 600-lawyer Nixon Peabody LLP told its lawyers about a new compensation system last week that set starting salaries at $120,000 to $190,000, depending on the market. It said Friday it doesn’t plan to make any further changes.

Only a sliver of law-school graduates each year land at firms paying the highest salaries, which dwarf entry-level pay in other industries. New psychologists and CPAs are paid less than $60,000 a year, according to consulting firm Mercer LLC, which also said that entry-level professors earn $80,000, and physician assistants, $110,000.

Law firms typically increase the hourly rates billed to clients every year, which, paired with the steady flow of work, will make the salary increases easy to absorb, law firm leaders and consultants said Friday.

Large law firms charge clients between $371 and $665 an hour for lawyers straight out of law school, according to Bodhala, a market-intelligence firm. Rates for lawyers in their third year of practice jump to between $473 and $914, Bodhala data show.

Law firm leaders say they are in a fight for top talent as higher-than-usual attrition rates during the pandemic leave them needing even more lawyers to churn out work.

There are more than 9,000 job listings for lawyer and staff roles posted on law firm websites, according to legal intelligence firm Leopard Solutions, more than it has ever tracked before.

“Money alone may not sway many in this market,” Laura Leopard, the firm’s founder, said. “Work-from-home flexibility and other perks will be needed to win in this atmosphere.”

Write to Sara Randazzo at sara.randazzo@wsj.com