The short answer: May, June, or the first half of July.
The longer answer is a trade-off between cape quality, daylight, weather windows and bull behaviour.
Late April. The very start of the rut. Bulls are dropping into the nanny basins. Coats are dark and thickening; capes are passable but not at their winter peak. Daylight is reasonable, around eleven hours. Weather is often more settled than mid-winter — autumn bluebird days happen. A good early-season window if you want to hunt before snowline drops.
May. The classic month. Daylight around ten hours, cold mornings, snowline dropping, bulls actively in the rut. Coat is good. Most outfitters' first trophy weeks of the year run in May. Expect at least one weather day per week.
June. Deep autumn into winter. Daylight around nine hours. Coat is at its peak. Snowline is often at or below 1,800 metres. Multi-day fronts more likely. Sub-zero nights at camp. Bull behaviour is at its best — full rut, bulls bold on open faces, big groups visible at glassing range. June is when the legendary photographs get taken. It is also when the legendary epics happen — big snow, blocked passes, helicopters grounded.
Early July. The trailing edge of the trophy window. Capes still good. Weather is harshest. Daylight short. Most outfitters wind down their trophy season by the second week of July.
August through April. Off-season for trophies. Capes are too thin. Population control work, nanny culls, second-tier hunts are all viable.
A few practical points on weather. The Southern Alps sit in the path of relentless weather systems coming off the Tasman Sea. South-westerly fronts can drop the snowline a thousand metres in a few hours. Sustained southerlies are common from May through September. Above 1,800 metres a winter front can bring sub-twenty windchill and whiteout. Settled "high" days exist between fronts and are spectacular — bluebird sky, no wind, every face on the hill bathed in sunlight, every animal visible.
The simple rule: build weather days into your trip. A six-day hunt with one weather day baked in is more comfortable than a five-day hunt where your only spare day got eaten by the third front. Pack as if you will be wet for at least one full day. Trust the forecast — MetService Mountain Forecasts and the YR.no models for the Southern Alps are both reliable enough to plan stalks against.