The next cold front is crossing the southeast on Monday, followed by another one immediately behind it, later Monday into Tuesday.
The trajectory of the first one has it sliding across the southeast, bringing limited rain.
The second one has more of a peaking trajectory, and that will bring better rain - but not huge falls. It's a bit of rain for our farmers, and snow for the alps - but nothing like what we saw with the long weekend system.
All up, most areas that do well in a west/southwest airflow on the mainland, should see 5 to 10 mm, locally up to 20mm. The totals taper off as you head inland/eastwards.
Northern and western Tasmania are in a better spot for the path of these, set for 10 to 25 mm, locally up to 50mm. But again, it tapers off as it heads to the other side of the ranges, and Tasmania's southeast, including Hobart, is only likely to see up to 5mm.

It's already 'properly chilly' over alpine areas, and the second cold front brings another burst of cold air.
That means there is no 'warm up' ahead of this weather system, and all of it falls as snow in higher alpine areas.
The snow level/thickness lines show that snow should fall in most alpine areas. The darker blue indicates falls to around 1500 metres above sea level (most snow resorts are above to well above this elevation). This colder air moves eastwards after this timestamp:

Much of the Victorian alps should see snow down to around 1300 metres, while the 1100 metre line moves into Tasmania on Tuesday morning.
LOOKING AHEAD
For later in the week and into the weekend, conditions are quite dry over the southeast as a high moves through. Frosty nights return, but it doesn't look quite as severe as last week (as this weather system isn't as cold and the high quickly moves to the east). It may actually turn 'warm for June' in the afternoon on Friday and Saturday.
The next weather system is likely to begin next Sunday/Monday. Some models have been liking a big rain and alpine snowfall, but it is too far out far any confidence on that yet.