Tropical storm Jose is following Hurricane Irma into the Caribbean and its wind speed is also predicted to reach hurricane levels.
The US National Hurricane Centre said it believed the storm's effects could be felt in the northernmost parts of the Lesser Antilles island chain by early next week.
There was up to a 20 per cent likelihood of tropical storm-strength winds, defined as faster than 39mph, battering areas already under hurricane warnings due to Irma within the next five days.
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On Wednesday morning its maximum sustained wind speeds had reached almost 60mph and it was moving west, from its position some 1,200 miles east of the Lesser Antilles, at about 13mph.
Jose is expected to become a hurricane by Wednesday night.
The storm is predicted to move northwest, apparently missing the Dominican Republic, Puerto Rico, Haiti and Cuba. By Monday, it will be out into open ocean and still some way east of the Turks and Caicos islands.
The NHC said Irma ranked as one of the five most powerful Atlantic hurricanes during the past 80 years and the strongest Atlantic basin storm ever outside the Caribbean Sea and Gulf of Mexico.
It churned across northern Caribbean islands on Wednesday with a potentially catastrophic mix of fierce winds, surf and rain, en route to a possible Florida landfall at the weekend.
Irma is expected to become the second powerful storm to thrash the US mainland in as many weeks but its precise trajectory remained uncertain. Hurricane Harvey killed more than 60 people and caused damaged estimated as high as $180bn (£136bn) when it hit Texas late last month.
Additional reporting by agencies
