The annual rate slowed to 3.3 percent from 4 percent, the lowest since June 2021.
But it remains outside of the Reserve Bank's target band, dampening hopes of an early cut to interest rates.
The biggest contributors to the quarterly result were higher insurance, rents, rates and energy costs, which offset cheaper accommodation, recreational services and some electronic equipment.
The rise in non-tradable items - core domestic services - was 0.9 percent for the quarter, compared to a fall of 0.5 percent for imported goods and services.
"Insurance prices increased 14 percent annually to June 2024 - nearly double what we saw 15 years ago in June 2009, which was the previous highest peak in the series," senior manager of prices Nicola Growden said.
On an annual basis, inflation household costs rose 4.4 percent, miscellaneous costs such as insurance were up 7 percent, and alcohol and tobacco up 6.9 percent.
Closer to RBNZ target
The Reserve Bank (RBNZ) expected inflation to fall back into its 1-3 percent target zone this year, and when it is convinced it will stay there, only then will rate cuts begin.
In its latest monetary review the RBNZ appeared to soften its stance, acknowledging inflation pressures and expectations were easing in the face of a weakening economy, but interest rates would need to stay "restrictive" and any cuts would depend on the economic data.
The next key numbers for the RBNZ would be employment and wages numbers in early August, with the central bank looking for an easing in labour costs and signs of more slack in the market.
The latest data followed inflation easing to its lowest level in nearly three years in the three months ended March.
