Tropical Storm Erika hit Dominica, in the eastern Caribbean, causing floods and mudslides that have set the country back 20 years, its prime minister said.
Haiti and the Dominican Republic are now expecting 53mph (85km/h) winds.
A state of emergency has been declared in the US state of Florida, where the storm is expected on Sunday.
Dominica's Prime Minister Roosevelt Skerrit said in a televised address late on Friday that hundreds of homes, bridges and roads had been destroyed.
He said: "It is with heavy heart that I address you, you can well imagine the hell that it has been for me since I heard of the passing of Tropical Storm Erika and the damage it has done to our dear people and beloved country. But we all have to pull ourselves together.
"The extent of the devastation is monumental. We have, in essence, to rebuild Dominica."
Erika dumped 38cm (15in) of rain.
At least 31 people on the island of 72,000 people have been reported missing, according to officials with the Barbados-based Caribbean Disaster Emergency Response Agency.
Among the houses lost in the mudslides was that of 46-year-old security guard Peter Julian, who had joined friends after leaving work.
"When I returned, I saw that my house that I have lived in for over 20 years was gone,'' he told Associated Press.
"I have lost everything and now have to start all over again."
Other Caribbean nations have issued tropical storm warnings.
In Puerto Rico, Erika knocked out power to more than 200,000 people and caused more than $16m (Â£10m) of damage to crops including plantain, bananas and coffee, AP reported.
The US National Hurricane Centre said the system was expected to move north across the island of Hispaniola - shared by Haiti and the Dominican Republic - where the high mountains would weaken it to a tropical depression on Saturday.
Meanwhile, Florida Governor Rick Scott declared a state of emergency as a precaution with officials urging people to prepare by stockpiling food and water and fuelling their vehicles.
Separately, in the Pacific Ocean, Hurricane Ignacio strengthened into a hurricane with maximum sustained winds of 90mph, centred south-east of Hilo in Hawaii, and moving north-west.
A tropical storm heading across the Caribbean has killed at least 20 people on the island of Dominica, with dozens more missing.