He is on remand after appearing in the Auckland District Court on Monday charged with Millane's murder.
He will next appear in the High Court at Auckland on January 23.
Grace Millane's body will be taken home by her father David this weekend. Photograph supplied Millane was murdered on the weekend of her 22nd birthday, just a day after she arrived in Auckland as part of a one-year solo OE.
She had landed in New Zealand 10 days earlier and had been travelling around the North Island before settling in Auckland for a visit.
The 26-year-old man was charged with killing Millane a day before her body was found .
He applied for name suppression when he appeared in court on Monday, but Judge Evangelos Thomas refused to grant the order.
The man's lawyer then indicated he would appeal that decision in the High Court.
This meant the accused had to be granted interim suppression for the 20 working days allowed for him to make that appeal.
As of yesterday, no appeal had been received by the High Court.
Millane's body was found in bush in the Waitakere Ranges on Sunday afternoon, a day before her alleged killer appeared in court.
Following a forensic scene examination and post-mortem examination, her body was released to her family yesterday.
Police searching the bush near where Grace Millane's body was found. Photo / Doug Sherring Her father David Millane has been in Auckland for over a week, helping police with the investigation.
He was joined by his brother.
The siblings are expected to travel back to the UK with Millane's body this weekend.
The police investigation into her murder continues.
Yesterday a different relative of the accused, who also cannot be identified for legal reasons, posted a lengthy statement online addressing the charge and allegation.
It followed an earlier statement published several days earlier demanding people stop trying to make contact.
The relative stated they had been overwhelmed with people messaging and asking questions after hearing the news of the arrest.
Grace Millane was killed while on the New Zealand leg of her year-long OE. Photograph supplied Yesterday the relative's statement said it would be the only time they would publicly address the issue.
They said they were sorry for the Millane family's loss and that she didn't deserve what happened to her.
They added that the whole of New Zealand and especially the accused's family were broken-hearted.
The relative said the support they had received was appreciated.
After they learned of the arrest they said they had hardly slept and had bawled their eyes out.
The relative said if the allegations against the accused were proven to be true, they were despicable, atrocious and disgusting.
The statement ended with a tribute to Millane, the relative stating she would forever be in their hearts.
Police are still appealing to the public to help them find a shovel they believe is connected to Millane's death.
This week they released a photograph of a similar "Atlas Trade" hardwood long handle, round mouth shovel.
Detectives want to hear from anyone who may have found the shovel any time after Monday, December 3.
CAN YOU HELP? If you have information that could help the Operation Gourami team in the investigation into the murder of Grace Millane, please call 0800 676 255.
To pass on information anonymously, call Crimestoppers on 0800 555 111.
Police are still searching for a shovel in relation to the inquiry. 
