Inflation eased further in May, but some of the most persistent price increases slamming the budgets of families and businesses look like they could stick around for a while still.

The fight to tame prices has come a long way since last year, when inflation peaked at 40-year highs, reaching 9.1 percent on a year-over-year basis in June 2022. As the Federal Reserve struggled to rein prices in, experts feared that the central bank would have to move so aggressively to slow the economy that it would cause a recession.

But the latest snapshot shows a different picture - even if it is not one of pure success. A report Tuesday from the Bureau of Labor Statistics showed that inflation eased for the 11th straight month in May. Prices rose 4 percent in comparison with a year earlier, the smallest 12-month increase since March 2021 and an improvement from the 4.9 percent annual rate in April. Prices also rose 0.1 percent in May over the previous month.

But look a little more closely, and the report offers less comfort. "Core inflation," a narrower measure that strips out more-volatile categories, such as food and energy, is not letting up. That raises the risk that some underlying sources of inflation are becoming a permanent feature of the economy and could require the Fed to push harder to vanquish high prices.

"The bigger question for inflation is: Where is it going? Where does it settle out?" said Peter Boockvar, the chief investment officer at the Bleakley Financial Group. "Are we just going to go back to this 1 to 2 percent inflation trend that we got so used to? Or is there something so structural that after the spike, after the comedown, are we going to settle at 3 [percent]?"

He added, "That plays into: How high do rates stay, and for how long?"

Major stock indexes flashed green throughout the day. The Dow Jones Industrial Average closed up 145.79 points, or 0.43 percent. The S&P 500 index climbed 0.69 percent and the Nasdaq 0.83 percent.

Housing costs continue to be a big driver of overall inflation. Rent rose 0.5 percent in May over the month before, only a minor improvement from a 0.6 percent increase in April. Rental costs were still up 8.7 percent from a year earlier.

Costs for used cars and trucks also increased 4.4 percent in May in comparison with April, as they did the month before. Wholesale costs for used cars have been rising, and those increases are showing up in retail prices.

Core inflation rose 0.4 percent in May, as it did in April and March. Normally, the prices of core goods may rise by between 0.1 and 0.2 percent. Fed officials are especially focused on this particular measure since it helps them gauge the underlying sources of inflation that can be hardest to tame.

Wendy Edelberg, the director of the Hamilton Project and a former chief economist at the Congressional Budget Office, said the continued price increases show that people are still spending, but that could force the Fed to slow the economy more, even if central bankers skip a rate hike this week.

"If we have any hope of getting inflation down, goods prices have to fall, just outright fall," Edelberg said. "I expected that to happen many months ago. We had a couple of months of good news on that front, but then real consumer spending remained really strong, to my surprise, and perhaps to retailers' surprise."

Encouraging signs were sprinkled throughout the report. The category for household furnishings fell 0.6 percent over the month, marking that index's first decline since June 2021 and its largest one-month decline since August 2009. Airfares also decreased 3 percent over the month after a 2.6 percent decline in April.

To get inflation under control, the Federal Reserve has raised its benchmark interest rate at a breakneck pace since March 2022. Those moves have brought the central bank rate, the federal funds rate, to between 5 and 5.25 percent - the highest level in 16 years. The goal is for steep borrowing costs to curb demand for all kinds of lending and investments, including mortgages and auto loans, so that demand for new houses or cars can fall into better balance with supply.

In Charleston, S.C., managers at Neal Brothers are not expecting supply chains, prices or the overall economy to return to what they were. The export, packing and distribution services company is focused instead on adjusting to the "new normal" on products including diesel, liquid propane and timber, said Vice President Harry Griffin.

Griffin also worries that the Fed's fight to contain prices will hamper his business. Neal Brothers stores building materials and other goods that eventually will be used by contractors building homes. If home building stops because of higher interest rates, that could hit Griffin's bottom line.

"As long as these interest rates stay really high, it could affect the home building industry long-term, and that may keep importers from importing as much lumber products in as they have been," Griffin said. "It's all connected."

Much of the economy, though, has remained resilient throughout the Fed's aggressive fight against inflation. Employers added 339,000 jobs in May, the 29th straight month of strong job growth. The country does not appear to be barreling toward a recession. And while there are signs that Americans are spending less on restaurants, hotels and flights, that could help the Fed's attempts to curb prices in service industries, which have been especially susceptible to labor shortages.

President Biden touted Tuesday's report, pointing to steps his administration has taken to tackle the cost of gas, prescription drugs and health insurance premiums.

"At the same time, the unemployment rate has remained below 4 percent for the longest stretch in more than 50 years, helping to support wage gains over the last year, even after accounting for inflation," Biden said in a statement. "More Americans are in the workforce than in decades."

For much of the past 15 months, the Fed has rushed to catch up to inflation, often hiking the federal funds rate in big jumps. And it has always signaled that more work remains. But when central bankers gather for their June policy meeting Tuesday and Wednesday, their agenda will be somewhat different.

The widely held expectation is that policymakers will leave rates unchanged this week to give themselves some time to see how the past year's increases are filtering through the economy. Rate increases operate with a lag, and many economists argue that the drop in inflation over the past year has been driven largely by improvements in supply chains, the return of gas prices to normal levels and the gradual fading of the economic aftershocks of Russia's invasion of Ukraine. That could mean the full toll of higher rates has yet to be felt.

Still, significant sources of inflation have not been tamed by the Fed's moves. The housing market slowed as mortgage rates soared. But rent, which makes up a large share of the consumer price index, continues to be a major driver of overall inflation.

Igor Popov, chief economist at Apartment List, sees relief coming in the rental pipeline. Rent growth peaked about a year ago, Popov said, and he expects rent inflation to slow down much faster moving forward.

About 1 million multifamily rental units are slated to come online later this year and next, Popov said. Some of those new units will serve the higher end of the market, but the total influx will create breathing room for renters in general - and the overall inflation picture.

"The Fed has not been worried about the shelter component for some time because this has been expected," Popov said. "There's always uncertainty baked in. The top line [rent inflation figure] is still very high . . . but it has turned a corner."

Officials have not definitively said they have finished raising rates, and incoming data on inflation, jobs and consumer spending will help them decide whether to make further increases in the coming months. Also significant will be information on bank lending, which has moderated since a recent shock to the financial system made lenders more skittish about issuing credit.

"A decision to hold our policy rate constant at a coming meeting should not be interpreted to mean that we have reached the peak rate for this cycle," Fed governor Philip Jefferson said in a recent speech. "Indeed, skipping a rate hike at a coming meeting would allow the committee to see more data before making decisions about the extent of additional policy firming." (Jefferson's remarks carry extra weight since he was nominated to the Fed's No. 2 role last month.)

In Baltimore, Postman Plus Perry Hall is being hit from all sides. The pack-and-ship store has seen the cost of 250-foot rolls of bubble wrap nearly double. Transportation and shipping costs go up every few months, even since gas prices eased from the summer's peaks. All of the store's employees already start at $15 an hour, but the hot labor market means they could earn higher pay elsewhere.

Owner Sharon Greenbeck says she has tried to absorb as much of the cost as possible. But looking at her small business, Greenbeck said it's nearing time for her to pass higher prices on to her customers. She worries about how they will react. Already, customers raise an eyebrow if they want to send gifts to friends and the shipping ends up costing as much as the gifts. Greenbeck said she has even seen inflation encroach on the "little things." For instance, the rubber ducks she would have in stock for children have doubled in price.

"I sell less because people say, 'Oh, I'm not spending $2 on a duck,'" Greenbeck said. "So then my merchandise sits. It's an ugly cycle."