The U.S. job market still has some life in it yet, as new data reveals an unexpected May bump in job postings and hiring.

Why it matters: Signs have mounted in recent weeks of a cooldown in the U.S. job market — good news for the inflation outlook but would be bad news if it gathered momentum and went too far. The new data assuages that fear.

Driving the news: Employers had 221,000 more job openings in May than a month earlier, the Labor Department said this morning, partially rebounding from an April drop. The openings rate ticked up to 4.9% of total employment, from 4.8%.

• Employers also hired 141,000 more workers in May than in April. The number of people quitting their jobs or being laid off barely changed.

Between the lines: The uptick in job openings and hiring suggests that the labor market cooldown remains very gradual and uneven — and that there's something of a floor for labor demand and, therefore, incomes and growth.

• "The report was another sign that the labor market is holding firm," wrote Robert Frick, corporate economist with Navy Federal Credit Union, in a note.
• "So far there are no indications that job growth will flag this year, so consumer spending power will continue to increase and the expansion looks solid."

Yes, but: Even after the modest May bounce, job openings and hiring are still well below their rates of a year ago, supporting the notion that the labor market has moved decisively away from its overheated 2022 state.

• "You can see the labor market cooling off," Powell said on a panel in Sintra, speaking moments before the new data was released.
• "It's doing just what you would want it to do: cooling over time — not quickly, not steeply," Powell said.

What's next: June employment numbers are due out at 8:30am ET Friday. Analysts surveyed by Bloomberg anticipate the report will show a deceleration in job growth to 189,000 positions added (compared to 272,000 in May).

• The consensus forecast is for the unemployment to be unchanged at 4%.

Powell said he is not focused on the possibility that former President Trump might be re-elected and could chip away at the Fed's political independence.

• "I do think support for Fed independence is very high where it really matters on Capitol Hill, in both political parties," Powell said today in Sintra. "I worry about getting the job right. That's what I worry about."

Why it matters: When Trump was in office, he publicly lambasted Powell, whom he appointed — once suggesting the central bank chief was a bigger "enemy" than China's Xi Jinping.

• Some in the Trump camp advocate aggressive steps to curtail the Fed's historical independence.

What they're saying: Powell pointed to the solid economy where the jobless rate is at a historically low 4%, growth is steady and inflation has been cooling.

• "I think if we can keep things on that path, that's what I am focused on. That's the only thing I am focused on," Powell said.
• On fiscal policy, Powell warned about America's big deficits at a time when the economy is at full employment.
• "This is something that should be a top-level issue. It should be a real focus going forward," Powell said. (He would not comment on the possibility of tax cuts if Republicans control both houses of government after 2024.)

Powell spoke alongside ECB president Christine Lagarde and Roberto Campos Neto, president of Brazil's central bank.

• Campos Neto is facing attacks from Brazil's president, who wants interest rates to be low.

The big picture: On inflation, Powell said the latest data suggests the nation is back on a disinflationary path.

Yes, but: "We want to be more confident before we start the process of loosening policy," Powell said.

The intrigue: "We are well aware that if we go too soon, that we can undo the good work we've done in bringing down inflation. And if we go too late, we could unnecessarily undermine the recovery and expansion."

• "We're aware we have two-sided risks now, more so than we did a year ago. That's a big change."