Jackson, 29, and his 26-year-old partner had been in the process of adopting the baby, who was called by the names Kadylak and Anna, police Lt. Todd Ojile said.
Jackson was booked on suspicion of child endangerment after the baby died Thursday in the Dodge Charger, where she was left for two hours on a 90-degree day. The other man, who was in the back yard at the time, wasn't detained because he believed all the children were safely with Jackson, Ojile said.
Officials said a decision on whether to file state charges against Jackson, who has two adopted children and four foster kids with his partner, would be made next week.
"The child had been picked up from the babysitter somewhere around 4 o'clock in the afternoon ... and somehow forgotten inside of the vehicle," Ojile said.
After two hours had passed, Jackson saw something on television that made him remember the baby, ran outside, got her out of the car seat and brought her into the home.
Police got a 911 call about a child who was unconscious and not breathing, and they were on the scene within minutes. The baby was pronounced dead at 6:49 p.m. (7:49 p.m. ET).
Ojile said it was still unclear why the baby was left in the car and why her absence went unnoticed for so long by her foster parents. He said both men were "extremely upset" when police arrived.
Kadylak's biological grandmother, Cindy Poe, drove from Topeka to Wichita to find out what happened.
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"Accidents do happen. We don't know if it was accident," she told NBC station KSN. "They have a lot of kids they take care of. I want answers."
Jackson's mother, Dottie, said he was inconsolable when she spoke to him at 2 a.m. Friday.
"He wants to die," she said. "Seth's children — that's his life."
She said he and his partner had adopted two boys, ages 5 and 7, who were taken into protective custody. They were fostering the baby and her sisters, ages 3 and 5, who were with other relatives at the time of the tragedy. The couple also has an 18-year-old foster son.
"I want Seth to know we're here for him," Dottie said, adding that she realizes his life will never be the same.
"His heart will break every time he holds a baby," she said. "It's going to break every time he walks in there and sees the crib."
The baby was placed with Jackson and his partner by a contracting firm that lost a competitive bid to renew its license last year but which was still allowed to sponsor licensed foster homes as a subcontractor.
The state Children and Families has opened an investigation and instructed two other licensed contractors to inspect all foster homes associated with the firm, Secretary Phyllis Gilmore said. In the meantime, all future placements with the firm have been suspended.
Neighbors described Jackson and his partner as doting parents.
"They are two of the most kind-hearted guys that I have ever met. And I hate that there's so much controversy right now with babies' being left in the car, because I truly don't feel from the bottom of my heart they would ever do this on purpose," said Lindey TenEyck, who lives across the street.
A mother of two, TenEyck realized something was amiss when fire trucks, police cars and ambulances converged on her street.
When she went outside, Jackson was on the ground near the silver Dodge Charger, practically in the fetal position.
"He was mentally breaking down. He was hysterical. There is no doubt in my mind he will suffer for the rest of his life in his mind," she said.
George Dasis said he broke down in tears when police carried the baby, bundled in a blanket, out of the house two hours after they were called.
"I feel very sorry for the guy," he said. "I don't think he meant to do it. He probably just wasn't thinking."
More than 36 children die in overheated cars every year in the U.S., research shows.
After the Wichita tragedy, the group Kids and Cars renewed its call for new technology in cars that could prevent other deaths.
"The fact is that our vehicles already remind us to buckle our seat belts, warn us if our gas tank is getting low, let us know if the keys are left in the ignition, or if a door is open," it said in a statement.
"With all of these reminder systems already in place, including a warning if our headlights are left on, who has decided that it's more important not to have a dead car battery than a dead baby?"