CODRINGTON, Antigua and Barbuda — Paul De Windt abandoned his home and holed up in his office with his family and pet cat on the island of St. Martin to wait out the fury of Hurricane Irma.

Days later, his house is uninhabitable, at least 10 people have died on the island, water and power have been knocked out, and half of the homes have lost their roofs.

But instead of thinking about the long and painful recovery to come, Mr. De Windt was still in his office on Friday, bracing for yet another catastrophe: the next major hurricane was already on its way.