Diana Nyad has already been stung several times by jellyfish but is continuing her 166km (103 mile) journey, team members said.
She left the Cuban capital, Havana, on Saturday afternoon and is aiming to reach Florida on Tuesday.
She is relying on an electronic shield as a shark deterrent.
She had to bring the attempt forward by 16 hours because of bad weather forecast for late Tuesday.
The long-distance swimmer was forced to abort her most recent attempt last September after potentially deadly jellyfish stings. An earlier attempt was meanwhile cut short by shoulder pain and an asthma attack.
On her blog, her support team wrote that after 19 hours of swimming, Nyad had progressed more than 22 statute miles towards the Florida shore and was still maintaining her regular pace of 50 strokes per minute.
They said she appeared “very comfortable and confident”.
She had spent a difficult night with multiple jellyfish stings – on her lips, forehead, hands, and neck – some of them from box jellyfish, they added.
To minimise the threat from the creatures, she now has a special head-to-toe swimming suit to wear at night.
She also has added jellyfish experts and detectors to her support team – alongside divers, whose task is to protect the swimmer from sharks, reports the BBC’s Sarah Rainsford in Havana.
If that fails, a team of divers is on constant alert to plunge in and guide sharks away themselves, our correspondent says.
Before the swim, Nyad said she felt “excited” but that she was aware of the dangers.
“It’s not perfect, you can see the white caps (on the waves). But I feel strong and… hopefully it will be calm so we can make it across before the big east wind starts coming in.
“I feel really excited. I respect this. I know how difficult it is – there is a reason no-one’s ever done it. But I’m prepared. I may suffer some, but I’m prepared for that too.”
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