Sir Patrick Vallance, Britain's chief scientific adviser, said "there is evidence that there is an increased risk for those who have the new variant".
The evidence about how much worse the new variant is was "not yet strong" and more research was needed, he said.
He said for a man in his 60s with the original virus, "the average risk is that for 1000 people who got infected, roughly 10 would be expected to unfortunately die" but "with the new variant, for 1000 people infected, roughly 13 or 14 people might be expected to die".
As of late last week, there had been 29 cases at New Zealand's border of the new British variant.
The requirement for pre-flight testing has been widened to countries beyond the United Kingdom and United States.
Workers at the border and in quarantine facilities will begin to be able to voluntarily get daily saliva tests as well as their weekly regular tests.
Otago University health professor Nick Wilson said: "We really do need a serious look at reducing the number of infected people arriving, and improving the quality of the whole border control arrangements.
"There's a genuine case for actually stopping all arrivals [from Britain] until we improve the border facilities."
Overseas there has been a renewed focus on top-quality masks as the best way to starve the virus of opportunities to spread.
Nursing leaders in Britain want a review of whether their equipment gives them enough protection against new variants.
The Royal College of Nursing said it was aware some National Health Service trusts are using higher-quality masks in their hospitals, while others use standard masks.
Germany and France have urged people to upgrade to medical-grade masks.
Chancellor Angela Merkel and German state leaders agreed last week that either surgical masks or respirators should be worn in the workplace, on public transport and in shops.
In France, masks are compulsory in all indoor public spaces, as well as on the streets in about 400 towns and cities. Government advice is that only a mask that fully covers the nose and mouth must be worn.
Double-masking - wearing a cloth covering over a surgical mask - improves protection close to the level of medical N95 respirators. Senior members of the new Biden administration in the US have been photographed and filmed using two masks at once.
It appears to have been a worldwide lost opportunity that there hasn't by now - after an entire year - been a flood of top-quality masks manufactured specifically for the public. Cloth masks are of varying quality.
Wilson said better masks needed to be worn by border staff and returnees in the facilities.
He also suggested that ventilation be improved and an argument could be made for more rural facilities to house returnees, rather than hotels.
In the continued absence of vaccines, other parts of the border operation need to be carefully assessed to gain any advantage.