MESSINA, Sicily — The old music school is overflowing. The building is now a migrant shelter for unaccompanied minors, able to house 224 teenagers — except that it is overstuffed with 281. And it is only early April. Migrant season in Sicily is just beginning.

Ninety miles away near the ancient hilltop town of Mineo, the story is the same. At a repurposed military housing complex that is now Sicily’s biggest center for migrants, nearly 1,000 new people arrived in March. They were among the almost 10,000 migrants rescued at sea last month and brought to Italy, a fourfold increase from a year ago. Add to that tally 2,150 people rescued at sea just this Monday and Tuesday.

Now that the European Union has struck a deal with Turkey to curb the record refugee flow into Greece, the question is whether the migrant flow into the Continent has been stanched, or whether migrants will simply find a different entry point, such as the more dangerous sea route from Libya into Italy. Migration into Greece has dropped sharply since the new deal, and it is too soon to know if the Syrians, Iraqis and Afghans who favored that route will shift their attentions to Italy.