"We appreciate the many tributes, words of support, and prayers we have received over the past 24 hours. Aroha nui," they posted to Twitter.
Aucklanders also took to social media to share images of the tower in tribute to Constable Hunt.
Constable Matthew Dennis Hunt was only 28 when he died during what was meant to be a routine traffic stop in Massey, Auckland.
Hunt started with police as a member of Wing 312 in October 2017.
He spent most of his career working frontline at Orewa and Helensville Stations before recently moving to work in the Waitematā Road Policing Team based at Harbour Bridge Station.
Hunt was raised on the Hibiscus Coast by his mother Diane, with his sister Eleanor.
He attended Orewa College.
"It was his life-long dream to be a police officer," his family said in a statement.
Before joining the police, he completed a BA in criminology and worked at Auckland Prison as a case manager.
He also spent time living in the United Kingdom before he returned to New Zealand to join the police.
"Matthew was a person of great integrity," his family said.
"His closest friends were like his brothers and sisters and they along with his family are absolutely heartbroken by what has happened.
"He was passionate about sport and his physical fitness and was thrilled to enjoy the recent Blues game at Eden Park with his mates."
