"The rescue team were dropped at the hut and then descended from there to the location of the injured man as the helicopter became grounded due to poor weather."
Rescuers carried Mr Rait for two hours before reaching the helicopter. He was pronounced dead by paramedics on arrival at Mt Cook Village.
Police said the spot Mr Rait fell from was only a few hundred metres away from the Tasman Saddle Hutt.
Englishman Robert Buckley, 31, has been identified as the second climber who died in the Aoraki Mt Cook National Park.
Mr Buckley, who was living in Christchurch, was with three other young men who had attempted to climb the Sefton Bivvy from Mt Cook Village on Saturday.
The party appeared to be "insufficiently" equipped and inexperienced for the climb they had chosen, police said.
"They had hired some climbing equipment early that day and had managed to get within 80 meters of the bivvy. At this stage ... one of the party has slipped on the ice and has fallen some 700 metres to his death."
The other men alerted emergency services but were unable to move from their location.
When a helicopter and rescue team reached the party, they were not able to rescue the men and recover the body because it was getting dark.
As a result, the survivors spent Saturday night where they were and were picked up at first light.
Mr Buckley's body was recovered later in the day.
Both accidents have been referred to the Coroner, police said.
