Franklin County Sheriff Gary Toelke said authorities had some good leads in their search for the attacker and the baby, Abby Woods, but "nothing has headed us in a certain direction."
The child's mother said she did not know her attacker, police said. Stephanie Ochsenbine, 21, was in stable condition Saturday following surgery.
The FBI, Missouri State Highway Patrol and the Missouri National Guard were helping with the investigation, Toelke said.
The sheriff asked the public to report any woman who had talked about wanting a child or acted pregnant recently.
"The description is not locked in concrete," Toelke said. "It could be someone who had a child die recently or could not have children and has told people they were pregnant and needed to steal a child so their lie would not be found out."
Franklin County Maj. Mike Copeland noted that a sign celebrating Abby's birth was displayed in the front yard and was plainly visible from a highway.
The attacker had gained entry to the family's home Friday afternoon by asking to use Ochsenbine's telephone. Ochsenbine's boyfriend, James Woods, who is Abby's father, was at work at the time, police said.
They fought after the woman told Ochsenbine "I'm here to take your child," and Ochsenbine was stabbed several times and her throat was slashed.
Ochsenbine's 1-year-old son, Connor, also was in the house but was unharmed.
Ochsenbine passed out and police aren't sure how long she was unconscious.
When she regained consciousness, she picked up Connor and went for help, trying two houses where no one was home before arriving at neighbor Pat Bearden's home.
"She was standing there with blood all over her and said 'Somebody stole my baby,'" said Pat Bearden, 69.
"I'm just praying they find the abductor and get the baby back," Bearden said. "We're afraid to let our little granddaughter ride her bike now."
Ochsenbine made "quite extraordinary" efforts to get help, Toelke said.
"Considering she just had a child and considering her wounds, I'd say she's a pretty determined woman," Toelke said.
While no one has been ruled out as being involved, Ochsenbine and Woods are not considered prime suspects, Toelke said. "We're not focusing on them," he said.
Lonedell is a small town about 45 miles south of St. Louis.