They have not ruled out the possibility the mother-of-four was already dead when the car struck her.
The death has been referred to the coroner.
Schmidt was in New Zealand with her partner, Richard Konarski, when she died.
Konarski has been charged with male assaults female as a result of an alleged altercation before she died, but claims there was nothing sinister about her death, Fairfax reported.
Schmidt's mother, Rae Kelly, said Konarski "abandoned" her on the night she died.
She said police had asked her not to speak publicly about her daughter's death, but she was frustrated at not having answers almost a year later.
Kelly told Fairfax that Schmidt and Konarski went out for dinner in Bluff the night she died.
As they drove back to Invercargill the couple had an argument, resulting in Schmidt getting out of the car and taking off down the road.
She said Konarski looked for her daughter for an hour and then drove back to their accommodation in Invercargill, returning to the area at 6am the next day to find the road cordoned off.
Police told him they were investigating a fatality.
Kelly was angry with Konarski for leaving her daughter alone in a place where she had no contacts "and no way of helping herself".
"He abandoned her that night," she told Fairfax.
It has also been revealed that Konarski is facing a charge of assaulting Schmidt the night she died.
He is due to appear in the Invercargill District Court on September 13.
Konarski refused to speak to Fairfax about the specifics of the charge or what happened the night Schmidt was killed saying he did not want to "impede" the investigation or court process.
But, he said it was a "tragic set of circumstances" that "can be explained and will be explained in due course".
Konarski said he was still struggling to cope with the loss of his partner.
"She was the love of my life and we were supposed to be on the holiday of a lifetime," he said.
After her death Schmidt's son, Jack Caughey, told the Otago Daily Times that his mother's death was "crushing".
"Part of me was taken away and I can't get it back," he said.
"She was always kind and she was always happy. She would make sure she did anything she could to make people happy.
"She was beautiful - she didn't deserve this," he said.