The conditions on White Mountain were very good on June 3rd, pretty much snow-free until about 13,000 feet. Higher than that, some snow patches I would guess to be 50 feet long at most are present but are definitely not a problem since the trail switchbacks are not steep and the patches can easily be circumvented if needed. They were hard enough on the way up to hold your weight without slipping and soft enough on the descent to provide some pleasant cushioning. Under those conditions, you could climb the peak in tennis shoes... In addition, there were very few people on the mountain (we only saw 5 people total), which I found surprising for a week-end. You can judge for yourself with the following pictures.
First clear view of White Mountain:
Looking up the summit ridge:
Looking down the same ridge from the summit: