Forecasters predict some areas will be hit by up to 4in of snow and temperatures in Glasgow and west and central Scotland will then fall below freezing.
The Met Office has issued yellow "be prepared" weather warning for those areas.
A spokesman for forecaster MeteoGroup said: "A further weather system will move in from the west and there will be a second band of snow covering the Central Belt and pushing north from early on Monday. We could see about 1in falling in some areas and 2-4in in the mountains.
"Most of the country will have some level of snow cover over the next couple of days, with significant accumulations in many parts.
"Temperatures will remain low for most of the week."
North Lanarkshire, West Lothian and the Borders were among the areas that got snow yesterday.
Scottish Transport Minister Keith Brown said councils had increased the amounts of grit and salt to treat major routes.
But the AA warned that 75% of drivers were not prepared for conditions on the roads, and urged drivers to carry an essential winter kit and check their cars before getting behind the wheel.
The RAC is expecting up to 56,000 breakdowns across the UK. It has extra patrols on standby to help stranded motorists and expects call-outs to rise 20% or more.