Don't remember which year exactly I was there. But perhaps the approach I took is worth mentioning:
Daytrip from the Leiteralm cable car, above Algund/Vellau -- following the "Meraner Höhenweg" to Hochganghaus, then up to Hochgang and via the lakes and Milchseenscharte to Halsljoch. Then standard normal route. Returned same way.
Yep, that approach to the Halsljoch is a possibility.
The Spronser lakes that you passed can be reached from several directions. From Schutzhaus Hochgang to the south, as you did, from the Bocker Hütte to the east, or from the Pfelderer valley to the north. The latter is what I described in the trip suggestion at the bottom of the Routes section, and I added that there are many variations. I think it would go too far to go into details.
On the map the route from Schutzhaus Hochgang to the Halsljoch looks short, but as you'll surely remember it takes quite a bit of time, because you have to gain a lot of altitude along the way, first up to the Hochgangschartl and later to the Milchscharte. Furthermore, the trails are rough, especially up the Milchscharte. If I get around to writing a page for Tschigat, I'll certainly include the approach from Schutzhaus Hochgang. But for the Lazinser Rötelspitze I don't consider it an important one, because it is much longer than from the Lodner Hütte and similar to what it takes from the Pfelderer valley.
Earlier in the day, I had already climbed the Lodner. That ascent went faster than I expected and so I decided to find another way down there than the normal route. That worked out fine, and I ended up descending the Andelsferner route to Andelsboden, from where I would have to cross the Halsljoch to get back to the Lodner Hütte where I had started the day.
Although I was already tired by the time I got to the Halsljoch, I figured that I still had enough energy to scramble up the SE ridge. After all, it's only a little more than 200 m up. Then, on the summit, feeling experimental once again, I decided not to return the same way, but try to find a way down the western slopes instead. There was a small risk that, somewhere on the descent, I would find no way down anymore and have to go back up again, but I was optimistic: I had already had a good look at the western slopes yesterday, during my hike up the Zieltal to the Lodner Hütte. The slopes didn't look real steep, so I figured it was worth trying, and it proved to be easy.