* Headline inflation eases to 4.24 percent vs poll 4.41 percent * Consumer prices dip 0.05 percent in early June * Jobless rate falls in May despite slowing growth MEXICO CITY, June 24 (Reuters) - Mexican inflation cooled in early June as fresh food prices fell although price gains are set to remain above the central bank's 4 percent ceiling for the fourth straight month. Inflation in the 12 months to mid-June eased to 4.24 percent from 4.63 percent in the year through May, below the 4.41 percent expected in a Reuters poll, data released on Monday showed. The central bank, which targets inflation of 3 percent, had expected inflation to start to turn down this month and keep trending down for the rest of the year, partly due to weak growth in Latin America's number two economy. But no further interest rate cuts are expected after a surprise decrease in March to the current 4 percent, with policymakers watching for the impact of an expected easing in stimulus from the U.S. Federal Reserve, which may weaken the peso further and push up import prices. Mexico's peso has fallen almost 4 percent since Fed Chairman Ben Bernanke said policymakers might tone down bond purchases later this year and stop them completely by mid-2014, if the U.S. economy performs as expected, and Mexican bond prices have jumped. Mexico is closely linked to the United States and growth is slowing in the first part of this year, although officials expect a recovery in the second half of 2013. The national statistics agency said consumer prices fell 0.05 percent in the month, compared to expectations for a 0.11 percent rise. Core consumer prices, which strip out some volatile food and energy prices, rose 0.07 percent, compared to forecasts for a rise of 0.12 percent. Separate figures showed that unemployment eased in May despite the weakness in growth, with the headline jobless rate falling back to 4.93 percent, from 5.04 percent in April.