Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Chiang Mai to Hamilton as you normally would.
For a flight from Chiang Mai to Hamilton, you will be flying from Chiang Mai. Chiang Mai (CNX) is just 3.1 km from the centre of the city. Chiang Mai is the only Chiang Mai airport and it handles 0 outbound flights to Hamilton per day. 0 airlines have flights to Hamilton from Chiang Mai on a regular basis.
This is defined as Chiang Mai Airport (CNX) and Auckland Intl Airport (AKL). Codes like Auckland Intl Airport are often used to just call airports uniquely. Tauranga Airport (TRG), Hamilton Airport (HLZ), Rotorua Airport (ROT), Taupo Airport (TUO), Whitianga Airport (WTZ) are some additional IATA codes for airports related to your trip.