The idea was to find a hike for 3 nights accessible by public transportation and with different start to end point so that it fully makes sense. There are several variations possible and I try to give an overview, day wise it's possible to lengthen it or to shorten it by one day and skip the last peak.

source: mapy.cz; here the editable version.
Access car
If you want to make the tour like described here, maybe best parking in St. Johann and taking bus to either Hüttschlag or Jägersee and going back with Railjet

Access public transportation
Train to St. Johann im Pongau (1h from Salzburg), from there bus to either Hüttschlag Talende (Start) or Jägerse (5).

Hüttschlag Talende

Hike
Day 1: via Salzburg main station to either Kleinarl Jägersee (point 5 on the map) or Hüttschlag Talende (start). We went to Hüttschlag and from there via the Kreealmen (2) up the 4h to the Albert-Biwak (3), it is just an emergency hut (8 Lager) and planned stays are not allowed but as the forecast was bad for the second day we wanted to shorten the hike for that day and took the Biwak.

After the Kreealmen the first time the glaciers of the Hohen Tauern get visible and via the Murtörl we proceeded towards the Albert-Biwak.

The Albert Biwak inside and outside, basic but awesome.

Alternatively you can start relaxed from Jägersee (5) and have a first night around the beautiful Tappenkarsee (4) at the Tappenkarseehütte and continue the next day with a very long stage to Osnabrücker Hütte (11h), bit before the half there is the Albert-Biwak (3) again so if you think you will not make the whole stage time/condition wise rather stay there and make two shorter stages.

Day 2: we started from the Albert-Biwak and went towards the Kölnbreinsperre storage lake (via Weinschnabel 2754m and in early July still shitloads of snow, if you're not used to hike through snow, just do this later in summer) and continued then 1.5h along the lake to the Osnabrücker Hütte (6).

Kölnbreinspeicher, to the left the dam and the hotel, to the right we followed the lake till the end to the Osnabrücker Hütte.

The views from the terrace of the hut towards the Hochalmspitze and its glacier are just stunning. The hike took due to the snow and bad visibility longer than expected and we needed more than 7h.
For more luxurious stay you can also stay at the Sporthotel Malta at the storage lake and possibly do some tours from there and then continue with the tour.

View bit away from the hut, same perspective, one of the nicest from any hut terrace towards the remaining glacier of the Hochalmspitze.

Day 3: from Osnabrücker Hütte (6) you go towards the Großelendscharte and to Hannoverhaus.

In the right back the Hannoverhaus gets visible.

Options here are: with glacier equipment and experience going via the Schwarzhornsee and the Kleinelendkees glacier straight to Ankogel (7) and traversing it down to Hannoverhaus (8).
Or starting early and doing Ankogel on the way with stable weather, should take around 10-11h, depending as always on condition and speed.
From Hannoverhaus you can already descent either via walking (2.5h) or with the cable car (in 2021 it cost 21€) to the valley if you want to stay shorter. Just keep in mind that you shouldn't arrive too late as cable car and trains should still be working (or staying in Mallnitz).

Day 4: starting early from Hannoverhaus to Ankogel. Begin July still a lot of snow and at 7am some iron (crampons, Grödel) on your shoes were definitely a big help, for the way down it's just awesome because via the snow it's incredibly fast and saves so much time. At 3096m is the Kleiner Ankogel, the view is pretty much the same like from the main peak so if you have not felt safe up there rather leave it here and don't continue to the main peak, the path was okay but due to snow/mud some parts are slippery and slipping isn't a good option there. The part is graded as I difficulty and most of the time a okay path but a couple of times for two metres you need to avoid any mistakes.
View from Kleiner Ankogel to the main peak. The path goes pretty along the ridge, always bit the the right side.

The way back is just the same like coming up, back to Hannoverhaus and the options again either via cable car or via pedes down to Mallnitz.

From the cable car station there is regularly (every one to two hours) a bus going to Mallnitz-Obervellach Bahnhof. If the times are not suiting there is also a nice path along some meadows to Mallnitz center and the train station, 40min to Mallnitz and then another 10min to the train.
We used the time in Mallnitz and went for lunch to the Sport-Cafe, nicely located on the kind of main square and good food (typical Kärntner Nudeln, as well Kasspätzle or pizza).