McClain is charged with capital murder in the shooting death of Kala Marie Golden in Spring, Texas. Witnesses say McClain repeatedly shot Golden and drove away with her 3-day-old son, Keegan Schuchardt.
According to the arrest record, McClain admitted to shooting Golden and taking Keegan to her home in Harris County. The warrant says Keegan was then found there late Tuesday night.
McClain is now being held without bail.
Montgomery County District Attorney Brett Ligon told ABC's "Good Morning America" that McClain's statement to investigators indicates that she shot the mother as part of a wider plan to kidnap any child and that Golden was simply in the wrong place at the wrong time.
"There were statements as indicated in the arrest record that were made by Ms. McClain that led us to believe that, in fact, this was an intentional act on her part," Ligon said. "Not that Ms. Golden was targeted specifically, but that this was part of a plan to kidnap a child."
He made no mention of a man whom witnesses said they saw in the blood-splattered Lexus.
Golden was leaving the pediatric center after a checkup with her son around 2 p.m. when she and a woman (presumably McClain) in a Lexus parked next to her pickup truck began to argue, Montgomery County Sheriff's Lt. Dan Norris said.
Witnesses said the woman repeatedly shot Golden, took the infant and sped away, hitting the dying mother as she tried to reach into the Lexus screaming, "My baby!" Norris said.
Witnesses also reported hearing as many as seven gunshots and said a man was also in the sky blue or light green Lexus, which was blood-stained on the driver's side, Norris said.
Meanwhile, Golden's mother, Linda Golden, told the Associated Press Tuesday that she had been baby-sitting another of Kala's sons 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.
Linda Golden said her daughter died "trying to save her baby."