His son Trenton Hill said police are reviewing security camera footage and are looking for two people with a dog who last saw his father.
He added the duck pond at Harcourt Park was drained and drones have searched the area.
Police have confirmed they are following up information received from the public, but would not say what that information is.
Trenton Hill told RNZ it was his and his father’s birthday on Saturday – the day before he went missing.
Some of his whānau are coming over from Australia within the next week to assist with the search.
“We’ve all taken time off work, and we’ve all just recouped at the house, and we’re just planning what to do,” he said.
“Hopefully we’ll find my dad soon. I have high hopes, but obviously, it could be bad, but I won’t think about that right now.”
He said he has received more than 4000 messages since Sunday.
“We’ve heard a million things of where he could be.
“We’ve had sightings in Porirua, sightings in Trentham, sightings in Petone, sightings in Taranaki even.
“And it’s been four days now, so there’s every possibility that he could be anywhere in New Zealand right now.”
The 50-year-old was described as 195cm tall, of solid build, with black hair and a short, white beard.
Trenton previously told RNZ his dad had just got a new job and had been doing really well for the past three years. He said he had a past of mental health issues, but it had been a long time since he had “an episode”.
“It’s been three years, nearly four years since the last episode, so this coming out of nowhere, just doesn’t make sense.”
He said if anyone saw something of note, people could contact him via his Facebook profile “Trent Bone”.
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Search for missing Upper Hutt man Bret Hill continues: ‘He could be anywhere’
