It said the heaviest and most frequent snow showers were likely over northern and western Scotland and Northern Ireland.
However, there will also be gales across the east coast which could cause temporary blizzard conditions and trigger hail and thunder.
Snowstorms are expected to make travel “very tricky”.
The new warning came as winds gusting to 77mph halted ferries and hampered traffic on major bridges today.
Engineers were battling to restore power to some 600 properties in northern Scotland which were cut off.
Three days of yellow - “be aware” - severe weather warnings are in force until midnight on Friday.
Gusts of 77mph were recorded in Tiree, 73mph in South Uist and 70mph at Machrihanish on Kintyre.
Sailings were cancelled on 20 of CalMac’s 26 west coast routes, leaving Arran cut off and no sailings to or from Barra, Colonsay, Cumbrae, Gigha, Iona, Islay, Lismore, Mull, Skye and the Small Isles.
The Forth Road Bridge was closed after a lorry was blown across the carriageways.
The Erskine, Dornoch and Skye bridge were closed to high-sided vehicles such as lorries.
Met Office chief meteorologist Paul Gundersen said: “Most northern areas are very likely to see snow showers at times over the next few days.
“Daytime temperatures are likely to be around 2 to 5C, but the strong to gale force winds will make it feel much colder.
“Overnight, a frost and ice will develop in most places, with some severe frost likely where there is snow on the ground in the north.
“Temperatures could dip below -10C over snow cover in rural areas.”
Up to 20cm of snow is possible above 300m in the north and west, and 10cm in the south east of Scotland, with 2-5cm at lower levels.
Police Scotland warned of “difficult driving conditions” .
A spokeswoman said: “The strong winds will be the most immediate issue.
“When combined with the sleet and snow showers, there is also a risk of snow drifts on higher routes on Thursday and Friday.”
A Met Office spokeswoman warned: “Travel conditions will start to turn very tricky, resulting in longer travel times.
“Be aware of the potential for disruption to transport due to lying and blowing snow.”
Lightning could also cause power cuts.
Ice and hail threaten further problems tomorrow, and trains could be affected.
In the east, winds could reach 75mph over high ground across Fife, the Lothians and Borders today.
That will be followed by the risk of snow showers from noon tomorrow to noon on Friday.
Transport minister Humza Yousaf said: “We will be seeing cold and windy conditions over the next few days, and plans are in place to help us do all we can to mitigate the impact of the snow and gale-force winds.
“We would ask that people keep the conditions in mind when they are making their travel plans and leave plenty of time.”
Chief Superintendent Andy Edmonston, head of road policing for Police Scotland, said: “With a risk of weather-related disruption for the remainder of the working week, motorists should bear in mind the risk of potentially hazardous driving conditions to come.
“Please ensure your vehicle is well prepared before setting off, making sure your windscreens are completely free of snow and ice, ensure your number plate is clear and your lights are working and clean.
“Ensure you have sufficient fuel in the vehicle, as well as supplies for yourself and passengers, including food, water, warm clothes and a fully-charged mobile phone.
“Motorists should also adapt their driving to suit road conditions, as well as the behaviour of other road users, including leaving more space between vehicles.”