I've hiked a few marked long distance trails from start to finish, and hiked parts of many of them, but mostly I just go wherever I feel like. Often I don't even plan ahead more than a few days. The Austrian Alps have a dense network of trails, and a few are described (or marketed) as long distance trails. Though none are as famous as the TMB, there are plenty to choose from.
If you like to watch glaciers, the Berliner Höhenweg could be your thing. If you want to take it one step up, and actually cross them (for some stages you'll need glacier gear to do that), take a look at the
Ötztaler Trek - I suggest that you do part of it, not the whole thing. Or, for a really long trip, why not hike the
Via Alpina, which crosses many countries, including Austria.
You mention June. That's early in the season. Not all huts will be open, especially in the first half of the month. And there can be snow higher up on the trails where it would normally be gone later one. That means you've got to be flexible and adjust your route if necessary (or pack crampons and ice axe, which is what I usually do when I think there is a chance I might need them).
Cheers, Rob