Likely flight is Emirates 435 from Auckland to Brisbane
No FlightAware track, so using the FR24 one (attached), at 5:38 it was 100 miles NW of Manurewa, which looks about right for the perspective.
Viewed from the side there's a change in climb rate (so this might actually be an increased rate of climb, rather than level then climbing as I put in my first graphic)
Viewed end on you can see the jump looks more significant with perspective
Ground level view.

The track is going left to right whereas the contrail is right to left, but contrails often are blown away from the track, which can be greatly magnified by the perspective compression.