The east coast and northern isles have been particularly badly affected, with all schools in Shetland and 65 in Grampian closed.
Forecasters say that Fife, Perth and Kinross, Angus, Aberdeenshire and Moray and the Highlands will be the worst hit areas.
The latest warning, which was issued on Monday evening, comes after heavy snow and blizzards caused chaos across much of Scotland.
Roads were closed throughout the country and many people were left without electricity after blizzards brought down power lines.
The east coast and northern isles have been particularly badly affected, with all schools in Shetland and 65 in Grampian closed.
Roads in Aberdeenshire, the Borders, West Lothian and Tayside have been blocked, while police have been warning motorists not to travel unless it was absolutely necessary.
Efforts have been continuing to restore electricity supplies to 3,000 homes in Argyll, the north east and Tayside after power lines toppled in blizzard conditions.
A spokesman for Scottish Hydro Electric said crews were working to restore power to homes by Monday evening.
Customers affected included 900 homes on the Isle of Arran, 900 in Toward near Dunoon, and 500 in Banchory, Banff and Tayside respectively.
The company has released an emergency number and can be contacted on 0800 300 999.
Drivers across Scotland were warned of the treacherous driving conditions.
Grampian Police said the A96 Aberdeen to Inverness road was difficult in many locations - particularly in the Inverurie, Glens of Foundland, Huntly, Keith and Fochabers areas.
Some stretches were reduced to a single lane and were passable only with extreme care.
School closures
Drifting snow also affected parts of the A90 - both between Aberdeen and Dundee and also between Peterhead and Fraserburgh.
Police said the majority of minor and side roads throughout Aberdeenshire and Moray were badly affected and have advised motorists to delay travelling.
At Aberdeen Airport all flights were subject to delay and a number of flights were cancelled due to the weather.
Sixty primary schools and five secondary schools were forced to close in Grampian, including: Inverurie Academy, Mearns Academy, Alford Academy and Banchory.
A number have already said they will be closed on Tuesday as well.
Several schools in the Highlands also closed, with the worst-affected areas being Easter Ross and the Black Isle where six primaries shut for the day.
In Shetland all schools were closed and there was no public transport services. Police advised drivers to keep off the roads.
Lothian and Borders Police reported that the A701 was blocked at Moffat and the A697 closed between High Cross near Lauder and Carfraemill.
The A68 was also closed south of Carter Bar.
Most minor roads in the Borders were reported to be either blocked or best avoided.