The victim's condition was improving, Det Sgt Harvey said.
''He's come a long way in three days,'' he said.
''I believe they are now looking at moving him into the children's ward at Dunedin Hospital.''
The 9-year-old boy involved in the accident was still coming to terms with it.
''He's still pretty upset,'' Det Sgt Harvey said.
''The whole family is pretty upset.''
Police believed the stake came from outside the school, but a ''few final inquiries'' were needed to confirm its origin.
''At this stage, we are not 100%, but it's looking like it came from outside the school,'' he said.
Police would make a decision on its most likely origin within the next day or so, he said.
There were no other witnesses to the incident, which had presented some difficulties for police.
Board of trustees chairman Paul Brooks said the school had arranged support for the 9-year-old involved in the accident.
He could not comment on when either of the boys would return to school.
The accident had brought several issues into focus for the school, he said.
''We will be looking at a number of things, including our health and safety policy, which we are in the throes of doing right now.''
Ministry of Education head of sector enablement and support Katrina Casey said the ministry's trauma response team remained at the school yesterday.
''The team will be there to help as long as they are needed,'' she said.
''Whenever a school experiences a distressing incident, of whatever kind, we offer support of this nature.''
A WorkSafe NZ spokesman said WorkSafe was informed of the incident, but would not investigate, as it was not a matter for the workplace health and safety watchdog.