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Wife and I have spent all our vacations camping and hiking for 30+ years. We hike every day on vacations, in the 8-15 mile range. Have 4000+ miles of hiking in national parks alone.
Glacier WILL be mostly under snow, and meltwater (a.k.a. mud) in early June. If there is any way you can postpone the trip and go later, do so.
That said, the favorite trails for us (and recommended in books) are: Avalanch Lake, Grinnel Glacier, Highline Trial, Iceberg Lake, and Gunsight pass.
I don't know that you would be able to do these hikes without an ice axe.
By the way, have you verified when Going to the Sun Road is open --- do they have it opened by early June?
I would love to go there but I think that it would to cold, even though I live in Michigan I can't stand that kind of cold.