A warmer-than-expected inflation reading released Thursday offers new ammunition for Federal Reserve hawks who are arguing for a gradual pace of interest rate cuts.

The Consumer Price Index for the month of September was higher than economists anticipated, increasing 2.4% over the prior year in September. That was still a slight deceleration compared to August's 2.5% annual gain in prices.

On the Fed’s preferred "core" measure, which eliminates volatile food and energy prices, the index rose 3.3% compared with expectations of a 3.2% rise. That was up a tenth of a percent from August.

This report, according to some Fed watchers, is unlikely to change the path outlined by policymakers for smaller future cuts following an initial 50 basis point reduction in September.

Investors, in fact, boosted the odds that the Fed will trim its policy rate by 25 basis points in November to 87% following the CPI release.

Read more: What the Fed rate cut means for bank accounts, CDs, loans, and credit cards

But the warmer reading on inflation coupled with the stronger-than-expected September jobs report will likely strengthen the argument by hawks on the Fed’s interest rate setting committee that any future cuts should be gradual.

These voices include Fed governor Michelle Bowman, the lone dissenter on the September rate cut who wanted to move slower given her concerns about inflation.

Atlanta Fed president Raphael Bostic has also voiced lingering concerns. Bostic told The Wall Street Journal Thursday following the CPI release that he was “totally comfortable” with holding steady next month and that he had already penciled in an estimate of just one more rate cut this year.

"I think we have the ability to be patient and wait and let things play out a little longer … There are elements of today’s report which I think validate that view."

Omair Sharif, president of Inflation Insights, said the new CPI reading isn’t likely to change the Fed’s calculus, especially since there were signs that housing costs are moderating.

"The Fed still wants to progress slowly, so I think 25 is the base case for November."

Not everyone agrees. Eric Wallerstein, chief markets strategist with Yardeni Research, said this new information not only gives the Fed hawks “more sway in November” but that it’s now possible the Fed won’t cut again this year.

"I think the Fed stands pat for the rest of the year," he said, adding that the central bank’s pivot in September with a jumbo-sized cut "was certainly premature."

In fact, Fed minutes released on Wednesday revealed that a robust debate took place at that September meeting about whether to cut by 25 basis points or 50 basis points.

A "substantial majority" of Fed officials supported lowering rates by 50 basis points, according to the minutes, but "some" would have preferred to have lowered by 25.

What those those minutes indicated is that while a chunk of the committee favored 25 basis points, some may have been swayed by Fed Chair Jay Powell and others to go larger.

In the weeks since that meeting, many Fed officials have been publicly urging a gradual path to cuts as they look to balance downside risks to unemployment with the ongoing effort to bring down inflation.

New York Fed president John Williams became the latest official to voice that view on Thursday, saying in a speech, "I expect that it will be appropriate to continue the process of moving the stance of monetary policy to a more neutral setting over time."

Williams didn’t comment on Thursday’s CPI report, but he also didn’t veer from the latest Fed playbook of stating that officials are pleased with progress on inflation while still acknowledging more work to do.

"There’s still some distance to go to reach our goal of 2%, but we’re definitely moving in the right direction," he said.

One of his colleagues, Dallas Fed president Lorie Logan, said Wednesday that “a more gradual path back to a normal policy stance will likely be appropriate from here to best balance the risks to our dual-mandate goals.”

Powell also made it clear in remarks on Sept. 30 that the central bank isn’t in a "hurry" to bring interest rates down and would prefer smaller cuts.

He also reiterated that the consensus of Fed officials outlined at the September meeting was for two more 25 basis point rate cuts this year, saying, “It wouldn't mean more fifties.”

"We anticipate a 25 bps rate cut in November but stress that the economic picture will be muddied by recent hurricanes, strikes, and Election Day two days before the FOMC meeting," said EY chief economist Gregory Daco.

"Unless there is a notable weakening in underlying payroll growth, hours worked, and the unemployment rate next month, policy gradualism will likely prevail."

The next test is the next jobs report due out on Nov. 1, just days before the Fed's next policy meeting of Nov. 6-7.

"If inflation data continues to indicate that prices are generally rising amid a backdrop of a cooler labor market, the Fed's next meeting will undoubtedly involve a more heated discussion of which of the Fed's mandates takes precedence," said Quincy Krosby, chief global strategist for LPL Financial.

Chicago Fed president Austan Goolsbee told CNBC Thursday that there had been some "close-call-type" meetings at the Fed in recent months and "there will probably be more close call-type meetings."

Click here for in-depth analysis of the latest stock market news and events moving stock prices

Read the latest financial and business news from Yahoo Finance