WASHINGTON, June 24 (Reuters) - Inflation may be even stronger in coming months than Federal Reserve policymakers currently expect as the U.S. recovery likely gains steam in the fall and a global recovery follows, St. Louis Federal Reserve president James Bullard said on Thursday.

That could push the level of prices beyond what is needed to account for recent years of inflation below the Fed's 2% target, and presented a "new risk" that Fed officials will have to consider in coming months.

"Inflation may surprise still further to the upside as the reopening process continues, beyond the level necessary to simply make up for past misses to the low side,” Bullard said in a presentation prepared for delivery to the Clayton Chamber of Commerce near St. Louis.

Bullard earlier this week said he was among the Fed policymakers who expect interest rates will need to increase next year. Faster than expected inflation this year and on into 2022 will, he said, meet the Fed's intent to let the pace of price increases exceed the formal 2% target for a time to offset years in which inflation has been too low.

His comments Thursday suggest concerns of an even larger inflation shock.

There has already, he said, been a "substantial" surprise in terms of stronger than anticipated economic growth and inflation. As schools reopen in the U.S. in the fall and more countries reopen their economies, "the risk is tangible" inflation could move higher than projected.

“Policymakers will have to take this new risk into account in the months and quarters ahead,” he said.

(Reporting by Howard Schneider Editing by Chizu Nomiyama)