U may want to stay at Taupo for at least a night

1) the sunrise at Lake Taupo is stunningly beautiful.

2) the only way you can take a morning walk along the trail.

3) and have time left to do the Shotover Jet Boat ride.

You can put the night at Taupo before the night at Wellington.

Unless you have particular reason, otherwise I dont see why you needs to spend a night at Hamilton.