Hi there, thanks for your query.
In New Zealand, there are mountains that have snow all year long. In May, it is not usual for snow to fall to lower levels as it it late Autumn then, so you could strike it lucky then too.
"Most of New Zealand's mountains are in the South Island. The 23 peaks over 3,000 metres high are all in the Southern Alps, which stretch for 500 kilometres down the South Island. This is a breathtaking landscape of towering peaks, ice, snow and huge glaciers such as Franz Josef and Fox." - Source