Smith, 40, is thought to be in Brazil and authorities are working with South American officials to find him and bring him home.
He was jailed for life in 1996 with a minimum non-parole period of 13 years for murder and child sex offence charges.
Basil Smith said he didn't know why his stepson would go to Brazil or who had helped him obtain a passport and $10,200 in cash - which he declared at Customs.
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He did not know of any friends or relatives in the Auckland area that Smith could have asked to help.
But he knew of support people involved with his temporary releases - he had joined his stepson in Auckland on a recent release.
"The support person ... he seemed to get on with pretty well," Basil Smith said.
"He drove us all around Auckland, took us up the Sky Tower. We walked around, went shopping, had dinner, that sort of thing. We just chatted."
On his other releases Smith "just enjoyed himself".
"He went to see Billy T James' grave [at Taupiri] and was telling me all about that. He just did normal things, like shopping, walking around town and going for meals."
He said Smith was not a risk to the community.
"He's not dangerous, not at all. He's easy going and he'd do anything for you."
Basil Smith said his stepson's victims need not worry about him trying to find them - as he had done in the past.
Watch: Killer Smith's escape: How did it happen?
"Since he has been inside he has pulled himself together. He does a lot of courses, he's very intelligent. Right through primary school he never put his brain in action, but since he's been inside he has."
Basil Smith appealed to his stepson to do the right thing. "Just come home," he pleaded
A police officer who dealt with Smith when he lived in Carterton said he had always had ideas above his station. "He was well known to police (before the murder). He had some pretty grandiose ideas," said the officer, who would not be named.
"He had big ideas about being a pretty good con artist."
Smith was not known to be violent back then. "He wasn't dumb, he was very manipulative ... he was calculating."
Smith, who obtained a passport under his birth name Phillip John Traynor, yesterday contacted Auckland human rights lawyer Tony Ellis, who has been his lawyer since 2002.
Smith told him he would issue a press release giving his side of the story tomorrow.
Police Commissioner Mike Bush yesterday revealed that an international warrant for Smith's arrest, which would prevent him crossing borders, was issued only at lunchtime yesterday.
Police had to "go through a process", collate accurate information and put evidence before a judge, who then issued the warrant.
Brazilian police were assisting New Zealand authorities and a Washington-based police liaison officer was travelling to Brazil to liaise with local authorities.
He would not rule out the possibility Smith obtained more passports.
Ten senior police officers in Auckland were working on the case and nobody in New Zealand had been arrested yet, despite suspicion Smith had help in acquiring a passport and fleeing the country.
Internal Affairs Minister Peter Dunne said yesterday his department had used its powers under the Passport Act to recall Smith's passport and said his passport renewal form was filled out manually, meaning it is almost certain that an accomplice helped him with the application.
Mr Dunne defended his department, saying that because Smith applied for a passport in his birth name it did not throw up any red flags, but information-sharing between Internal Affairs and law enforcement would have to be reviewed.
The sister of Smith's murder victim said his plan to make a public statement stemmed from a desire to be famous and have his "day of glory".
"He always wanted to be famous. I guess he is right now," said Lynda, sister of the man Smith murdered in 1995.