The latest yellow - “be aware” - warning, for 6pm tonight until 11am tomorrow, will see up to another 10cm of snow fall, the agency said.
All 30 schools in Shetland were shut today, along with 13 in the Highlands, six in Moray and one in the Western Isles.
School buses to nine schools in Aberdeenshire were cancelled because of ice and snow on the roads.
Routes blocked by snow included the A939 Gairnshiel-Corgarff, B976 Crathie-Gairnshiel, B974 Banchory-Fettercairn, A941 Dufftown-Rhynie and B9002 Lumsden-Cabrach in Aberdeenshire, and the A9007 Carrbridge-Ferness in the Highlands.
The A702, which links Edinburgh with the M74, was closed by flooding at Coulter, south of Biggar.
On the railways, trains were delayed by snow blocking points at Tomatin, between Aviemore and Inverness, and a snowdrift at Banavie, just north of Fort William on the Mallaig line.
Dalwhinnie was one of the coldest places last night, where it dipped to -11C.
SCOTSMAN TABLET AND MOBILE APPS
The Met Office said further travel disruption could be caused by the next snowfall.
A spokesman said: “Showers of sleet and snow will continue to affect many eastern and particularly north eastern parts of the UK through the rest of Tuesday and into Wednesday giving icy stretches on untreated surfaces.
“Rain and sleet is more likely near exposed coasts, but slightly further inland local snow accumulations of 1-3cm are possible above around 100m, with perhaps 5-10cm over the Grampians.
“Most inland areas of the south should miss the snow. The snow showers will ease across much of Scotland later in the night.”
The Met Office forecast further wintry showers in the east on Thursday and said milder weather was not now expected until Saturday.