I can recommend a trip in the eastern part of the Hohe Tauern range at Ankogel Group. This area can easily be reached by train as the main route crosses the Tauern range in a tunnel between Badgastein / Boeckstein to Mallnitz
A glacier is only safe if it is free of snow. Even if there is no fresh snow in September, this will not be the case in higher regions. So you should concentrate on the ridges. I can recommend a trip like the following:
1. by train to Mallnitz, walk up to Arthur-von Schmid-Haus
2. climb Saeuleck and proceed to Hochalmspitze along Detmolder grat route. This requires good conditions. The final ascent to Hochalmspitze crosses the upper part of Trippkees glacier, but there are no crevasses if you stay above the bergschrund. Crampons and ice axe maybe necessary. Some parts have fixed cables like a via ferrata. Descent to Giessener hut. The alternative decent to Osnabruecker hut crosses Hochalmkees glacier what is too dangerous for going solo.
3. change huts from Giessener hut to Kattowitzer hut by descending into the Maltatal valley.
4. climb Grosser Hafner (almost the easternmost 3000m-peak of the alps in the morning and walk to Osnabruecker hut in the afternoon.
5. follow the Tauernhoehenweg trail to Grosselendscharte pass and to Hannoverhaus hut. Ankogel (3250m) can be climbed on a marked route (easy ridge) in addition
6. follow the Tauernhoehenweg trail to Hagener hut. From there Gesselkopf (2974m) can be climbed on marked route in addition.
7. descend to Nassfeld / Sportgastein and take the bus to Boeckstein or Badgastein (train station)
Of course variations are possible. E.g. on day 3 a direct marked route from Giessener hut to Hanoverhaus is possible and a prolongation of the route from hagener hut into Goldberg Group.
It is a very nice area. Have fun! In september the weather can be very stable at high pressure as there is no risk of thundrstorms. But if bad weather arrives you should consider changing your plans to goals more on the south side of the alps like Dolomites.
Online map availabe at: http://www.austrianmap.at/