At least one death was reported in Jamaica, where Melissa roared ashore on Tuesday with top sustained winds of 295 km/h.
A tree fell on a baby in the island's west, Abka Fitz-Henley, a state minister, told Nationwide News Network, a local radio station, adding that most of the destruction was concentrated in the southwest and northwest.
Flooding from Hurricane Melissa killed 25 people in nearby Haiti, officials said.
Jean Bertrand Subrème, mayor of the southern Haitian coastal town of Petit-Goâve, told the Associated Press that 25 people died after La Digue river burst its banks and flooded nearby homes.
Dozens of homes collapsed and people were still trapped under rubble as of Wednesday morning, he said.
Melissa had top sustained winds of 165 km/h and was moving north-northeast at 22 km/h on Wednesday, according to the National Hurricane Center in Miami.
The hurricane was centred 70km northwest of Guantánamo, Cuba, and 335km south of the central Bahamas.
Preparations for the storm in the Bahamas "should be rushed to completion," the agency said.
Hundreds of thousands of people in Cuba had relocated to shelters.
A hurricane warning was in effect for Granma, Santiago de Cuba, Guantánamo, Holguin and Las Tunas provinces as well as the southeastern and central Bahamas.
Melissa was forecast to continue weakening as it crossed Cuba but remain strong as it moves across the southeastern or central Bahamas later on Wednesday.
It is expected to make its way late on Thursday near or to the west of Bermuda.
Haiti and the Dominican Republic also braced for its effects.
The storm was expected to generate a surge of up to 3.6 metres in the region and drop up to 51cm of rain in parts of eastern Cuba.
Intense rain could cause life-threatening flooding with numerous landslides, US forecasters said.
The hurricane could worsen Cuba's severe economic crisis, which already has led to prolonged power blackouts, as well as fuel and food shortages.
"There will be a lot of work to do. We know there will be a lot of damage," Cuban President Miguel Díaz-Canel said in a televised address and urged the population not to underestimate the power of Melissa, "the strongest ever to hit national territory".
Jamaican officials reported complications in assessing the damage.
"There's a total communication blackout on that side," Richard Thompson, acting director general of Jamaica's Office of Disaster Preparedness and Emergency Management, told the Nationwide News Network.
More than half a million customers were without power late on Tuesday as officials reported that most of the island had downed trees, power lines and extensive flooding.
Extensive damage was reported in parts of Clarendon in the south and in the southwestern parish of St Elizabeth, which was "under water," said Desmond McKenzie, deputy chairman of Jamaica's Disaster Risk Management Council.
The storm damaged four hospitals and left one without power, forcing officials to relocate 75 patients, McKenzie said.
The United Kingdom said on Wednesday it was deploying 2.5 million pounds ($A5 million) in emergency humanitarian funding to assist the Caribbean region's recovery from Hurricane Melissa, with targeted support for Jamaica.
The aid package includes the rapid delivery of shelter kits, water filters and blankets to help prevent injury and disease outbreaks, the government said.
