The man accused of her murder, a 24-year-old from Mt Roskill, handed himself into police hours after her body was found. Police said he was not known to her.
His name and any information that may identify him is currently suppressed by the courts. His next appearance is later this month.
Auckland City Police Detective Inspector Kevin Hooper, the officer in charge of the Operation Solitaire murder inquiry, would not be drawn on details of the latest attacks.
"Investigations are ongoing and we are not in a position to say anything further."
Pert's father, Kevin Pert, said he was "horrified" to learn there may have been other attacks on women the day his daughter died.
"We are just absolutely horrified about the whole thing," he said.
"After what happened to my daughter ... I can't believe it."
The family was struggling to come to terms with her death. "We still can't believe it happened to us."
Friends and family have described Pert as a wonderful mum and an intelligent, driven woman. Her close friend was MP Paula Bennett, who had known Pert for 25 years.
Police have previously said that Operation Solitaire investigators were seeking information about the yellow Mazda Demio the accused was driving when he turned himself, in as they try to piece together his movements in the 24 hours before the alleged attack.
Earlier this week the Herald revealed the accused may have been spotted trying to break into houses the day before her death.
On Wednesday, police officers were door-knocking in Grey Lynn - 12km from where Pert was believed to be attacked.
Information was still being gathered from people who saw Pert out running that morning, or who were out running in the area themselves.
It is thought Pert left her Viaduct apartment around 9am for her run.
Police want to hear of sightings of a yellow Mazda Demio from January 6 and midday January 7. Operation Solitaire: 0800 765 842 or solitaire@police.govt.nz
