By Pete Schroeder and Howard Schneider

WASHINGTON/AMELIA ISLAND, Florida (Reuters) -Federal Reserve policymakers said on Tuesday the U.S. central bank should wait several more months to ensure that inflation really is back on track to its 2% target before cutting interest rates.

"In the absence of a significant weakening in the labor market, I need to see several more months of good inflation data before I would be comfortable supporting an easing in the stance of monetary policy," Fed Governor Christopher Waller told the Peterson Institute for International Economics in Washington.

The timeline was echoed by Cleveland Fed President Loretta Mester in comments on Tuesday night at an Atlanta Fed conference.

"I need to see a few more months of inflation data that looks like it is coming down," said Mester, who will attend her last Fed meeting in June before retiring.

"We're in a period when patience really matters," Boston Fed President Susan Collins said at the same event. "I think the data has been very mixed ... and it's going to take longer than I had previously thought."

The policymakers spoke as the Fed has downplayed talk of any further rate hikes, but also noted they feel the economy needs to cool further.

"We just don't want to go off a cliff. That's the critical thing," Waller said. "We are not seeing anything right now that looks like staying here for three or four months is going to cause the economy to go off a cliff." The comments were echoed by his colleagues who cited the low perceived risk of keeping rates elevated for now.

The Fed has kept its benchmark policy rate at 5.25%-5.50% since last July and, stung by three months of stronger-than-expected inflation readings from January to March, is only cautiously welcoming more recent encouraging signs of a loosening in the labor market and return to further progress in lowering inflation toward its 2% target.

"We read this as confirming Waller is open to cutting in September if, but only if, a more definitive downshift in inflation unfolds over the coming months," Evercore ISI Vice Chairman Krishna Guha said.

The remarks also appeared to add to market confidence in Fed rate cuts, with traders firming up expectations for a first reduction in borrowing costs in September and a second one at the Fed's final policy meeting of the year in December.

'RATHER WAIT LONGER'

In a separate appearance, Atlanta Fed President Raphael Bostic struck a similar tone to Waller, noting the central bank needs to be cautious about approving its first rate cut to be sure it does not touch off pent-up spending among businesses and households, and put policymakers in a position where inflation reaccelerates.

"It is in our interest to not start bouncing around. ... For me, I'd rather wait longer to make sure that doesn't happen," Bostic told reporters on the sidelines of the Atlanta Fed conference in Florida. He said he still expects inflation to edge lower through the year with a single rate cut being appropriate in the fourth quarter.

"I'm not in a hurry to cut rates," Bostic said. "We need to make sure that when we start on that path, it's unambiguous that inflation is going to get to 2%. ... The existence of this potential exuberance means that we have to be very cautious about when we do that first move, and it may mean that it has to happen later."

At a separate event at the Dallas Fed, Fed Vice Chair of Supervision Michael Barr added his voice to the chorus, repeating that higher inflation readings in the first quarter had not added to his confidence that prices pressures are easing.

"For me at least, that means we need to sit tight where we are for longer than we had previously thought," he said. "We need to see more evidence of continued progress on inflation for us to be in a position where we could think about adjusting the policy rate."

The policymakers spoke a day before the release of the minutes of the U.S. central bank's April 30-May 1 meeting, a key readout of a gathering that ended with officials feeling - in the words of Fed Chair Jerome Powell - that it "will take longer than previously expected" for them to have sufficient confidence in falling inflation to cut rates.