J.R. Kimbler, his 10-year-old son, Dakota, and his daughter, Jade, 6, appeared to be safe, the National Parks Service said. They were about two miles away from a trail in Congaree National Park.
Kimbler took his children on an outing Saturday. Tammy Ballard, the children's mother, said they were to spend the day with their father, but she learned at about 8:30 p.m. that Kimbler, a taxi driver, had sent a text message to a friend saying he was lost.
"I think that they just came intending on staying a couple of hours and then they got lost, because he had left his cigarettes and lighter in his cab and he's a heavy smoker," Ballard told WLTX-TV. "That makes me think he wasn't planning on being away from the car for very long."
Congaree National Park is a 27,000-acre expanse of old growth forest and swamp near Columbia, S.C. A large section of the park is a Wilderness Area.
The Park Service closed the park during the search.
Notable deaths of 2014 [PHOTOS]