“At present, participants judged that it was important to move expeditiously to a more neutral monetary policy stance. They also noted that a restrictive stance of policy may well become appropriate,” the minutes said.

This month’s half-point rate increase lifted the Fed’s benchmark rate to a range between 0.75% and 1%. Officials also unanimously approved a plan to begin shrinking the Fed’s $9 trillion portfolio on June 1 by allowing securities to mature without reinvesting their proceeds into new ones.

Fed Chairman

Jerome Powell

in recent public appearances had largely telegraphed the rate-setting committee’s intentions to raise rates by a half-percentage point, or 50 basis points, at its next two policy meetings, including in an interview with The Wall Street Journal last week.

“Most participants judged that 50 basis point increases in the target range would likely be appropriate at the next couple of meetings,” the minutes said.

Mr. Powell further signaled resolve to bring down prices by suggesting that the unemployment rate, at 3.6% in April, may need to rise as the Fed slows demand. “There could be some pain involved,” he said last week.

The minutes hinted more obliquely at a similar discussion at this month’s monetary-policy meeting. “Several participants commented on the challenges that monetary policy faced in restoring price stability while also maintaining strong labor market conditions,” the minutes said.

Until this month, the Fed hadn’t raised interest rates at consecutive policy meetings since 2006, and it hadn’t raised interest rates by a half percentage point since 2000.

In recent public comments, officials have appeared united on the need for half-point increases at the Fed’s June and July policy meetings. They have presented contrasting views about what should occur after that.

A few regional Fed presidents have said they would support pressing ahead with a more aggressive pace of rate increases in September if monthly inflation readings remain elevated.

“I will need to see several months of sustained downward monthly readings of inflation before I conclude that inflation has peaked,” Cleveland Fed President

Loretta Mester

said in a May 13 speech. If by September, “inflation has failed to moderate, then a faster pace of rate increases may be necessary.”

St. Louis Fed President

James Bullard

has called for the Fed to raise interest rates to around 3.5% this year, which could entail raising rates in half-point increments at every meeting this year.

But others, including

Patrick Harker

of Philadelphia and

Charles Evans

of Chicago, last week indicated optimism that the economy and inflation would slow so that the central bank could dial down its rate increases to the more traditional quarter-point increment in September. “This will be dependent on where the data are,” Mr. Harker said.

Atlanta Fed President

Raphael Bostic

told reporters on Monday that he could see a case for pausing rate increases in September. “Given the very high level of inflation, some might be surprised by my injecting some caution here,” he said in an essay published online Tuesday. “But remember this: even firetrucks with sirens blaring slow down at intersections lest they cause further preventable trouble.”

Mr. Powell last week laid out a relatively high bar to slow down rate increases. “This is not a time for tremendously nuanced readings of inflation,” he said. “We need to see inflation coming down in a convincing way. Until we do, we’ll keep going.”

Mr. Powell said he expected rates by the fourth quarter might reach a more neutral setting, but he said this wouldn’t warrant a pause in rate increases. “It’s not a stopping point. It’s not a ‘looking-around’ point,” he said.

The most recent inflation data has been mixed. On a monthly basis, the consumer-price index’s gauge of core prices—which excludes food and energy—rose a seasonally adjusted 0.6% in April, according to a Labor Department report last week, and increased 6.2% over the previous 12 months.

The Fed uses a different gauge: the personal-consumption expenditures price index. April inflation data from that Commerce Department report will be released on Friday.

(More to Come)

Write to Nick Timiraos at nick.timiraos@wsj.com