At least 12 shots were fired at the man shot and killed by police earlier this month, his family say.
Jerrim Toms, who was already facing charges of failing to stop for police and dangerous driving, was killed after allegedly approaching police with a machete after leading them on a long pursuit.
The 29-year-old's family told RNZ that police informed them that at least 12 shots were fired.
Four of the bullets were said to have hit Toms at close range, including one to his left hand, two in his chest, and one to his lower back.
They were fired by two police officers.
His family told RNZ they could not understand why so many shots were fired.
His father Kevin Toms said, "If [Jerrim] was rushing towards them and he was a direct threat to them he'd have to be real close.
"You don't miss two-thirds of your shots close up, so we can't tally that.
"He was running away, they'd spiked his car, he was frightened of them, he was out of there, he wasn't running towards them."
They also questioned why Tasers weren't used.
His family also told RNZ that Toms was treated for bipolar disorder and was a father-to-be.
Toms and his pregnant German partner were expecting their first child together.
Sister Natasha Toms said five weeks before the shooting, Toms had been hospitalised and treated for bipolar disorder.
The Herald understands Toms had been under the care of the Auckland District Health Board's mental health service. The ADHB refused to comment.
"At this time, we are referring any inquiries relating to this incident to the police," a spokeswoman said.
The family said Toms had not been "his normal self" since the occasion, and was on prescribed medication.
They said on the night of the shooting, Toms had relapsed and family had contacted mental health services and police for help.
Natasha told RNZ her brother had been portrayed as a criminal, but she believed he wasn't.
Information released in the days after the shooting said officers were called to an Onehunga address just after 3am on Saturday by a woman reporting that Toms was acting erratically.
She left the property, fearing for her safety.
Police arrived at 3.10am but Toms was not there.
Less than half an hour later, a police officer on his way home from work saw a car pulled over with its hazard lights on SH1 north of Auckland.
The car sped off when the police officer approached, and then stopped again.
The officer also stopped and the driver got out and approached the driver's side of the patrol car.
He had a machete and was threatening the officer.
The officer drove away from Toms and called for backup.
Toms then got back into his car and headed south on SH1 towards Puhoi.
More police, including the Eagle helicopter, were called to help.
Police said Toms was travelling at up to 150km/h and turned off his lights at times.
His car was stopped only after police put down road spikes at Pohuehue about 3.50am on Saturday.
As officers got out of their car to speak to Toms, he advanced on them, again armed with a machete.
Police shot at Toms and he was struck and killed.
New information has revealed that the knife Toms was welding at police was a blunt ornamental knife Toms had bought about seven years ago.
The family have said they believe recreational drugs may also have influenced his mental state on the night of the shooting.
Police have declined to answer questions about the specifics of the shooting, citing rules under the Coroner's Act and active investigations.