He was with his partner at the time, Michael Peter Atkins, now 54, who was later charged with murder but later acquitted by a Supreme Court jury in September 2009.
Detective Jubelin was later forced to reopen the case after police got a tip-off and there was increased pressure from the public to find his body.
He is now set to reveal how they put a "controversial deal" in place that led to finding Leveson's body in Sydney's Royal National Park on May 31 this year, 9 News reports.
His parents had never stopped looking for him, and were forced to dig for his body for nearly 10 years.
The Levesons have described the deal police made as a "deal with the devil", 9 News reports.
"How can you expect that someone who has lied for nine and a half years is about to tell the truth?," his mother Faye Leveson is seen saying on the 60 Minutes trailer.
The fresh investigation comes after 60 Minutes revealed CCTV footage of Atkins walking into Bunnings, and buying a mattock and duct tape at the cash register in 2015. Police had found a receipt for a mattock and tape in Leveson's car boot after he was reported missing.
Mrs Leveson left the Glebe Coroner's Court on Friday last week holding a photo in which her 20-year-old son Matthew is beaming.
The picture was taken two weeks before he died in 2007 and was secretly buried in bushland south of Sydney.
Faye's husband, Mark, on Friday displayed a much more recent photo of Matt to show the world what Atkins had done to their party-loving son. In this darker picture, most of Matthew's bones have been reassembled on a table, but his skull is in pieces and his hands and feet are missing. It's the last photo the Levesons have of their middle son.
It was taken after his remains were exhumed in early June.
"You don't leave someone you love like that," Mark Leveson told reporters outside the court.
Earlier, many in the courtroom cried when Faye spoke about her son's death. But the man who buried his body in the Royal National Park - Atkins - was not there to hear it.
"He should have been there to listen, to hear what we had to say," Faye said later outside court.
"Everybody in the court heard what he had to say but the one person who should've had to listen to what we had to say was him. He put Matt in the ground."
Mrs Leveson spoke with anger about Atkins, who last year agreed to lead police to his remains in exchange for immunity from perjury and contempt of court charges.
"If (Atkins) was truly genuine and loved Matt as he says, he would have told us where Matt was on September 23, 2007," she said.
"The only reason he gave us back Matt in May this year is because he was under threat of having to go to jail on perjury charges."
The news comes as Atkins will not be forced to reappear at an inquest into the death of his former young boyfriend when it resumes in Sydney.
NSW State Coroner Elaine Truscott on Friday last week discharged the subpoena that previously compelled Atkins, 54, to give evidence, saying he consistently lied and had "no credibility as a witness".
The inquest at Glebe Coroner's Court will resume on September 26 for final submissions.
