NEW YORK, May 8 (Reuters) - Strong U.S. hiring data in April dealt prospective Federal Reserve Chair Kevin Warsh's hopes of cutting interest rates a setback on Friday, giving officials greater latitude to use monetary policy to tackle increasingly high inflation.

The U.S. economy added 115,000 new jobs in April, exceeding analysts' forecasts, following an upwardly revised job gain of 185,000 in March. The April job gain was well above the amount of job creation many analysts say is needed to keep the job market steady. The unemployment rate held steady at 4.3%.

The hiring indicates the U.S. job market continues to do well at a time when inflation pressures are mounting on the back of the ongoing impact of President Donald Trump's import tax hike coupled with surging energy prices caused by the Iran war.

The data diminished what were already low odds that the Fed can cut interest rates later this year and strengthens the hand of the substantial number of Fed officials who are worried about inflation and want to hold rates steady for an extended period.

"The labor market is not booming, but it is proving harder to break than many feared," said Olu Sonola, head of U.S. economics at Fitch Ratings. "If unemployment stays this stable, the Fed's attention shifts back to inflation," Sonola said, adding that if price pressures remain robust "the Fed's easing bias is unlikely to survive much longer".

The market now expects just an 18% chance of an interest rate increase at the Fed policy meeting in December, according to the CME's FedWatch. It was at roughly 23% late Thursday. At the same time, the chances of the Fed keeping rates steady rose to 74.1%, compared with 70.1% the day before.

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The Federal Open Market Committee last week held its interest rate target range at 3.50%-3.75% and maintained a leaning towards further rate cuts in its policy statement.

That drove three regional Fed bank presidents to dissent, and in subsequent comments, they explained uncertainty over the outlook and the risks created by the Iran war mean it could even be possible the Fed would have to raise rates at some point.

Cleveland Fed President Beth Hammack said in a radio interview on Thursday, "I think our statement should ⁠have a pretty neutral stance about whether the next move is down or up or just on hold for a ​really long period of time".

The diminished odds of a Fed rate cut come as Warsh is zeroing in on Senate confirmation to succeed Jerome Powell, whose term ends on May 15.

Warsh has expressed interest in cutting interest rates but as of now, is likely to find few takers given mounting energy prices and an unresolved war that increases the odds price pressures will mount further the longer the conflict goes on.

Warsh may be further boxed in on the policy front as Powell intends to stay on in a governor slot that runs until 2028 while he seeks assurance Trump administration legal investigations targeting the central bank are over.

While Powell said "I'm not looking to be ... a high-profile dissident ​or anything like that" at the FOMC meeting press conference, his presence will likely strengthen the positions of those who oppose rate cuts.

In a TV interview Friday, Fed Governor Stephen Miran reiterated his case for cutting rates, arguing that the current monetary policy stance is likely holding the job market back.

Miran, who is serving after his term expired and will have to leave when Warsh is confirmed, also said he hopes Powell staying on is "transitional" and not a "nefarious" move that would muddy the waters of who is actually running the Fed.