During the storm that hit the South last week, 23 trains were damaged by the high winds and fallen trees.
South West Trains' staff have been working on the clear-up operation throughout Christmas, and so a full timetable will run for 94% of the network on 2 January 2014.
More than 140 trees were blown down onto the tracks by gales and a dozen trains were damaged by fallen trees.
In Hampshire, the line between Petersfield and Haslemere will remain closed until Monday 6 January after a major landslip in Liss.
A replacement bus service is running between Petersfield and Haslemere, with journey times extended by up to 30 minutes.
On the Isle of Wight, Island Line services remain suspended for the immediate future as work continues to repair severe track damage in 20 different locations near Ryde caused by flooding on Christmas Eve.
Tim Shoveller, Managing Director of the South West Trains-Network Rail Alliance said: "We are continuing to monitor the weather forecast and have been advised that more stormy conditions are likely over the next few days.
"Our priority will continue to be to make sure customers have the latest information to help them plan their journeys and to get people from A to B safely and as quickly as possible."
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Saturday's weather for the east of the region
Saturday's weather for the west of the region
Saturday morning's weather for the east of the region