The first named storm of the Atlantic hurricane season is posing an uncertain threat to the Gulf of Mexico, even as efforts to contain the worst oil spill in U.S. history are set to ramp up.

For now, Tropical Storm Alex, which is hitting the western Caribbean with rain and high winds, is not expected to pass close to BP Plc.’s blown-out well off the Louisiana coast.

But even a miss that only generates large waves could greatly complicate clean-up efforts from Louisiana to Florida.

Current official estimates suggest between 35,000 to 60,000 barrels a day are leaking from the rogue well. BP collected over 24,000 barrels on Friday and about 11,640 barrels in the first half of Saturday, the company estimated.

New equipment being moved to the site of the leak in the coming week could raise the daily collection rate to 53,000 barrels a day, U.S. Coast Guard Admiral Thad Allen, who is coordinating the U.S. oil spill response, said on Saturday.