The 43-year-old leaves behind his two daughters Emily, 7, and Olivia, 6, along with his loving wife Serena, the family member said.
The family planned to bring Smyth's body home from Waikato Hospital last night to be surrounded by whānau.
"We have yet to decide on the day of the service, and where he will be lying until that time. I will post again as soon as decisions have been made so you will know where to visit," a relative said in a post on Facebook.
"Thank you all for the lovely messages you have sent," she said.
Another simply posted: "Our hearts are broken", alongside photos of Smyth.
Yesterday, hundreds of Smyth's friends and family left comments paying tribute to him.
"Sean was the nicest, hardest-working and most genuine person I've ever met," one said.
"He told the most amazing stories and had the cheeky grin and unforgettable laugh to boot.
"He was loved by all and will be terribly missed. Like so many, I was honoured to be able to call him a mate."
Another wrote: "I work with Sean 's ex employer. They are devastated and have only spoken with absolute respect. Much love to your family at this hard time."
The Smyths are a well known family from Piopio.
Inframax Construction chief executive Chris Hayward said their focus today was still on looking after Smyth's family and supporting staff.
As for whether they were fully operational today, Hayward said they were not yet back to normal.
"Operations aren't functional today."
Emergency services were called to the yard of Inframax's head office in Waitete Rd about 11.40am on Monday.
Smyth's death has been referred to WorkSafe and the Coroner.
It comes after 16-year-old Shaldin Joyce died in a forklift incident last month in Ōtāhuhu.
On October, 4-year-old Jackson White died after being struck by a forklift on the family's work site in Canterbury.
On August 11 last year, a 63-year-old died after being crushed by steel framing that was transported by forklift, while in 2017 a 51-year-old died after being struck by a forklift in Canterbury.