Shetland has been issued with a Met Office amber warning for wind amid concerns that it might experience gusts of up to 100mph.
It comes after hurricane-force gusts left tens of thousands of homes without power across Scotland on Friday.
The storm caused the suspension of all ScotRail trains, although some limited services later resumed.
More than 28,000 homes are without power as the Atlantic jet stream caused gusts of more than 100mph (160km/h).
Engineers have restored supplies to more than 88,000 customers.
BBC weather forecaster Philip Avery said the Met Office warning for Shetland was in place from 04:00 GMT on Saturday until noon the same day.
Elsewhere, an amber warning for much of the rest of Scotland has now been lifted, but lesser yellow warnings remain in place across central and southern Scotland, northern England and Northern Ireland.
The yellow "be aware" warning of strong winds and a chance of snow is also in place for all areas of Scotland, Northern Ireland and much of England on Saturday and Sunday.
All ScotRail services were suspended while Network Rail, which is responsible for the rail infrastructure, inspected lines for damage caused by winds and high tides.
The rail operator said the Aberdeen Sleeper hit a tree at Cupar. The tree was cleared and the train was able to continue.
The majority of services remain suspended, but ScotRail said services between Glasgow Queen Street and Edinburgh Waverley have been restored on a reduced service.
Some other local services are also now running, with full updates available on the ScotRail website.
It was hoped most East Coast, Virgin and TransPennine Express services between Scotland and England would be able to run as normal, but passengers were warned there may be short notice alterations and cancellations. Some CrossCountry rail services have been affected.
Anyone planning to travel to or from a station in Scotland should check their journey in advance.
About 28,000 Scottish and Southern Energy (SSE) customers and 800 Scottish Power Energy Networks customers remain without power after widespread overnight disruption to the electricity network in areas of northern and western Scotland.
SSE had mobilised 1,000 technical and support staff ahead of the storm, with engineers working in "extremely challenging and potentially dangerous conditions" to reconnect electricity.
The company said it had restored supplies to 68,000 customers and was working to restore power to a further 28,000 customers.
SSE said they had managed to restore power to all homes in Orkney apart from Papa Westray.
About 1,200 engineers and support staff will be deployed on Saturday and SSE said it will be able to tell householders and businesses when they will get electricity back by 10:00.
Rural areas have been the worst hit, especially around Dingwall and in Inverness-shire, the Western Isles and Skye.
Other areas affected included parts of Aberdeenshire and rural areas around Wick, Oban and Fort William, as well as Buchan, Dunblane, Dunoon, Elgin and Huntly.
Scottish Power Energy Networks said it had managed to reconnect 20,000 homes, with a further 800 still affected by "pockets of faults mainly across the central belt, from Ayrshire to Lanarkshire and across to the Lothians".
The company said "the vast majority" of customers affected by the storm will have power restored by later tonight, "but this may depend on weather constraints and by the impact of current high winds causing additional faults".
Scottish Transport Minister Derek MacKay told BBC Radio Scotland that "safety has to be paramount when looking at the services that are operational" but said everything was being done to "get people moving".
Mr MacKay added: "Many travellers who have had sleepless nights will know the winds have been severe and yes it has caused significant impact to the transport system but we're working very hard to recover from that.
The Forth Road Bridge reopened to cars only at 07:25 after earlier closed at about 01:00 when a van overturned on the northbound carriageway as the wind reached 91mph. The bridge has since reopened to all vehicles.
The Dornoch Bridge and the Skye Bridge are closed to high sided vehicles, while a fallen tree closed the northbound carriageway of the Kessock Bridge.
Police said that travel conditions in the Highlands and Islands were "hazardous" and advised against travelling along causeways or low-lying coastal roads.
The Churchill Barriers in Orkney have been closed.
BBC News correspondent Colin Blane said police and fire services were dealing with many incidents of fallen trees.
The Scottish Environment Protection Agency has issued several flood alerts and flood warnings.
BBC Weather said gusts reached 113mph in Stornoway, 110mph at Loch Glascarnoch, and 97mph at Altnaharra. Speeds of 61mph have been recorded overnight at Glasgow and Edinburgh airports.
A gust of 140mph was recorded at the summit of Cairngorm and the BBC Winterwatch studio, in a cabin on the Mar Lodge Estate in the Cairngorms, was destroyed by the winds.
The yellow warning is also in place for northern coastal areas of Northern Ireland and parts of northern England, including Cumbria, Lancashire, Northumberland, Durham and North Yorkshire.
Cumbria Police said strong winds had blown over a lorry between junctions 38 and 39 on the northbound carriageway of the M6 motorway.
The Met Office's chief forecaster said a depression had been developing over the Atlantic in association with a very strong jet stream. It is said to have been triggered by sub-zero temperatures in the US hitting warmer air.
Some ferry services in Scotland have been cancelled.
All schools on Orkney and the Western Isles have been closed on Friday, with dozens of schools and nurseries in the Highland Council area also closed because of the weather.
Herriot Primary School in Renfrewshire has also been closed after the storm caused damage to its roof, and there have been a small number of closures in Moray and elsewhere.
Full details of school closures can be found on council websites.
The Dounreay nuclear complex on the far north Caithness coast will be closed to all but essential staff on Friday and Saturday, the site's operator has said.
The winds are predicted to die down on Friday before returning again overnight and into Saturday morning.
Snow showers are also possible across Scotland on Saturday.
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