Between 160 and 200 were on board the vessel when it sank, the US coast guard has said.
Rescuers on Wednesday found one more survivor on the nearby island of West Caicos, who was suffering from severe dehydration.
Authorities have so far rescued 119 passengers, 16 of them on West Caicos island. Most of the survivors were found sitting on reefs close the main islands.
|Rescued migrants were being taken to the Turks and Caicos islands
"[The accident] gives you an idea of the stakes that these migrants are willing to face in order to come to a place of greater economic prosperity," Nick Spicer, Al Jazeera's correspondent in Providenciales, the capital of the Turks and Caicos, said on Wednesday.
"The people who are still out there can survive in the water for a number of days - it is not cold water. This is a tourist destination and the water is relatively warm. But they do face sharks."
The US coast guard cutter Valiant is continuing to search the accident scene and a Falcon jet was launched early on Wednesday, Johnson said.
A series of searches will be held on Wednesday before the US coast guard, in consultation with Turks and Caicos Islands officials, will decide whether to suspend its search, she said.
"As with all people that need to be rescued at sea, the longer they are in the water, the less chances of survival they have," she said.
"They are dealing with exposure to the elements, and they have no food and water, or lifesaving equipment, which multiplies the trouble they could be in."
Some of the most severely injured survivors have been taken by helicopter for medical treatment to Providenciales.
The Turks and Caicos islands are a British territory in the Atlantic ocean, between the southern Bahamas and the north coast of Haiti.
Haitians often leave their country in crowded boats in an attempt to escape poverty and find work in the Bahamas or Florida.
"Haiti is the poorest country in the Americas and it has been pounded by hurricanes and bad weather of late. All of the efforts by the international community to improve the lot of Haitians have been hurt by the weather but also by the typical inertia that you find in the United Nations and the aid agencies," Spicer said.
"That is why Haitians hop on rickety boats to go across 1200 kilometres of ocean to the US in an attempt to improve their lives - because the aid to Haiti is not arriving fast enough."
Last week, the US coast guard found 124 Haitian migrants travelling on what they said was a "grossly overloaded" boat about 240 km southwest of the site of Monday's accident.
Those migrants were repatriated to Haiti on Monday.