There were a series of road accidents, including a gas tanker skidding off the A9 at Aviemore junction and a lorry owned by a charity jack-knifing nearby, close to Carrbridge.
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In Dundee, a car skidded off the road and into the side of a house after a six-car crash during rush hour.
No major injuries were reported.
Network Rail said its staff have been working “around the clock” to keep tracks clear.
Snow gates were closed at Braemar, Tomintoul, Cock Bridge and on the A82 between Tyndrum and Glencoe, while motorists around the country have been advised to drive with caution.
A total of 38 Highland Council schools and nurseries were closed, affecting 2,570 pupils.
National Express said there was “severe disruption to all services” due to the wintry conditions, while Stagecoach said city services were subject to delays of up to 50 minutes.
Police in the Borders reported a number of minor accidents due to the weather.
Heavy snow also hindered travel in parts of Argyll and a car was damaged when it crashed into railings on the icy slopes of Corran Brae, Oban.
Ferry services to the Northern Isles were disrupted, while a number serving the Western Isles were delayed or cancelled.
More gales, snow, hail, sleet, lightning and heavy rain are being forecast from today into the weekend, and possibly carrying into next week.
Met Office forecasters have upgraded their warning for snow to an amber “be prepared” alert across central Scotland, while yellow warnings remain in place for the rest of the country. The amber warning covers the Strathclyde, Central, Tayside, Fife, South-west Scotland and Lothian and Borders areas.
Forecasters said there was a risk of two spells of persistent snow, one overnight and the other in the morning, which could bring more travel disruption and difficult driving conditions.
The warning came as power was finally restored to homes affected by the fierce storms which began last Thursday, leaving some households without electricity for up to five days.
More than 120,000 properties were cut off as the first of gusts of up to 113mph brought down power lines.
With further bad weather forecast, the Met Office said: “A cold and very unstable westerly flow covers the UK. This will bring frequent showers of hail, sleet and snow overnight into Wednesday.
“Snowfall amounts will probably vary substantially across the amber area but there is the likelihood that some places will see very little settling snow.”
Some of the worst affected areas from the last week – including Orkney, Shetland and the Western Isles – look set to escape the worst of the weather.
Scotland’s transport minister Derek Mackay said: “Our trunk road operating companies are doing all they can to keep roads clear where possible.
“We have patrols out across the day and night and motorists can see details of where and when gritters will be in operation through the interactive map on the Traffic Scotland website.
“There is plenty of salt in stock and our winter control rooms are monitoring conditions 24/7 to make sure that the winter fleet is treating and patrolling where required.”
Chief Inspector Louise Blakelock, of Police Scotland, said: “With a substantial risk of disruption to travel likely due to the weather, motorists should consider whether their journey is absolutely necessary.”
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