The self-employed garden contractor told Fairfax he spent Boxing Day morning looking for his son in Nelson, in case he had managed to go further afield.
Hunt had left money and clothes out in case Alex returned home for supplies before leaving again.
"Even if he runs away again at least I'd know he had some money and a way to stay warm," he said to Fairfax.
Hunt has been searching town every night for Alex, putting up posters and appealing for information from the public.
Alex's mother Erika Petersen, who lives in Newcastle, Australia, told Fairfax she "just wanted her baby home" and was desperate to know if he was safe.
"The thought of Alex waking up on Christmas Day to nothing and nobody is unbearable," Petersen told Fairfax from Newcastle.
"He's my boy and I just need to know he's safe. He's still just a child.
"Someone's got to know where he is. Please just let us know he's safe."
Petersen told Fairfax her first instinct was to fly to New Zealand to help search for her son, however she had a newborn baby to care for.
"I can't sleep and all I think about is Alex," she said to Fairfax.
There had been no reported sightings of the 14-year-old, who left home wearing shorts and a T-shirt, but no shoes. Hunt told Fairfax Alex has run away before but only when he has been in some sort of trouble.
"I just want him to know he's not in any trouble and we just want him back. His family in Australia are all really concerned and his mum's worried sick.
"I just keep thinking why he would have left without any of his things, no clothes or shoes. He has a photo of his uncle who died and there's no way he would have deliberately left that behind," Hunt told Fairfax.
"I just hope he's safe somewhere and can let us know he's OK."
Alex was described as being of thin build with a shaved head. He is about 165cm tall.
Police have urged the public to report any possible sightings or to contact their local police station and quote file number 161220/3464.
Call Blenheim police on (03) 578 5279 or give information anonymously by calling Crimestoppers on 0800 555 111.