McClain's sister, Corina Jackson, told authorities that she had talked about needing to "do the adoption" soon after taking Keegan.
Her fiancé, who was not identified Wednesday, is being interviewed by authorities.
Officials said they did not believe anyone else was involved in the shooting and abduction.
Sheriff Tommy Gage said Kala Golden had placed Keegan into her pickup truck Tuesday afternoon after leaving Northwoods Pediatric Center in Spring, about 20 miles north of Houston. The suspected shooter was parked next to her, Gage said.
During the confrontation, the woman repeatedly shot Golden, then snatched the child from her truck and drove away, according to witness accounts. The dying woman leaned into the vehicle and tried to take the boy back, screaming, "My baby!" but her attacker sped off.
Two detectives spotted a vehicle later Tuesday outside a nearby apartment complex that matched witnesses' descriptions, Gage said. Though McClain's apartment was empty, she later arrived and talked to authorities.
"During her interview with detectives, information was obtained which led detectives to a residence in Harris County where her sister lives and a possible location of the child," Gage said. McClain was later arrested. Police say she admitted carrying out the attack.
"I don't understand that, her taking a child," her estranged husband, Theo McClain of San Diego, told The Associated Press. The couple had two children together and raised a third from a previous relationship of his.
"That doesn't make sense. We have three kids. And then to shoot somebody? I've never known her to hurt anybody."
Investigators said McClain's statements included information only the shooter would know and indicated she attacked the mother Tuesday as part of a wider plan to kidnap any child and that Golden was simply a convenient target.
She was being held without bail.
The baby has been returned to family members, according to Keith Schuchardt, who said he had been married to Kala Golden for three years.
He told the Houston Chronicle that he learned of the shooting because as Golden lay dying, she asked another woman to call him and explain the attack.
Asked by reporters what he would tell his wife now, Schuchardt said, "I wish you were here with me to get me through this."
Schuchardt told the AP that officials were initially concerned about his criminal record, which includes felony convictions for possession of a controlled substance and burglary of a coin-operated machine.
McClain is a vocational nurse, a job that involves providing basic nursing services under the direction of registered nurses and doctors. She does not work at the pediatric center, according to a clinic receptionist, Jackie Longoria.
McClain, who has vocational nursing licenses in both California and Texas, has not faced disciplinary action in either state, according to licensing boards.
Golden's mother, Linda Golden, told the AP that she had been baby-sitting when someone at the scene called her using her daughter's cellphone. She rushed to the clinic but was unable to see her daughter because paramedics were trying to save her.
"I wanted to kiss her before they put her in the ambulance," Linda Golden said.
Linda Golden said she had no idea what could have sparked the slaying and abduction.
"That's the hardest she's ever fought," she said. "She died trying to save her baby."
Associated Press Writer Danny Robbins in Dallas contributed to this report.