The King's Birthday big snow system has a little more to deliver on Monday.
Local winds are now southerly, but where they have come from is different. The low driving our weather has moved out to the southeast over the Tasman Sea. That brings in a little tropical flavour (as winds travel clockwise around a low) - so the air moving over the alps originated over the Pacific Ocean, rather than the cold Southern Ocean.

This injection does cause our airmass to warm a little on Monday - but it has great timing as the precipitation is easing.
In the week that follows there is the potential for a little snow but not a lot at this stage.
The next period of instability is on Thursday into Friday. This cold front brings a new burst of cold air (yay), but high pressure gets in the way and reduces the instability (booo). There is the potential for light snowfalls with this one.
We try again on Sunday into Monday. This one looks stronger on some weather models - but the key bit here is, it doesn't look strong on the majority of models. So, there is potential here, but low confidence at this stage.
The next one is on Monday into Tuesday next week. This one is shaping up to be more of a surge coming up from the southwest, and more likely to have a good mix of cold air and instability - but it is a full week away, and anything can happen between now and then :)