Due to multiple factors, expect to potentially pay more when booking in June. For low rates, November can be a great choice for travel to Hamilton International. Flights to Hamilton International from Dallas/Fort Worth Intl in November are about $453 on average during November, but can be found for as low as $269.
When travelling, it's generally smart to pack in preparation for rain or any other conditions that may arise. The rainfall in Dallas/Fort Worth Intl is roughly similar to Hamilton, New Zealand. You can assume this to be the case over the next 10 days.
You can imagine temperatures to be in the 36 °C range in July. The weather in Hamilton, New Zealand is generally colder than Dallas/Fort Worth Intl's. As flight availability is subject to become more limited as peak season approaches it is ideal to book in a timely manner.