Coach Ricki Herbert told reporters the All Whites would need to acclimatise as soon as possible for their Group F games against World Cup holders Italy, Paraguay and Slovakia.
The Oceania champions play Slovakia on June 15 in Rustenburg, Italy in Nelspruit on June 20 and their final group game against Paraguay in Polokwane on June 24.
All three venues are at altitude with Rustenburg 1500 metres above sea level, according to FIFA's World Cup website.
Polokwane is 1300m above sea level while Nelspruit is the lowest altitude at 600m above sea level.
"From a planning point of view we will look to move as quickly as possible now," Herbert told reporters on a conference call on Saturday, a day after the draw in Cape Town.
"Getting two European teams shapes our thinking a little bit and we may try to get some games in that part of the world prior to going, but we need to assemble now into South Africa a little bit sooner than what we thought (to acclimatise)."
Herbert said New Zealand Football (NZF) officials in South Africa for the draw would finalise training venues and accommodation on the high veldt and also arrange a series of friendlies over the next few days.
"Conversations get struck pretty quickly and it moves pretty fast," Herbert said, adding that they would ideally look at five or six matches in the lead-up to the tournament.
"I think it was just great to see New Zealand come out in the draw and get the country back on the world stage again ... (and) we go there with some confidence.
"If we get our planning and preparation right and the volume of games needed then I think we can go across there in a very positive way."
