The Met Office has issued yellow warnings for wind which could cause further power cuts, transport delays and damage to properties.
The wet and windy weather is set to roll in later today, with the weather warning covering Wales and most of England from midday until 3pm.
Wind speeds are expected to approach 60mph later this evening.
Identical wind warnings have been issued for Monday.
The warning comes as power providers said that Britain had suffered one of the biggest national outage on record.
Some 155,000 people are still without power, and the Energy Networks Association said it believes the UK may have experienced a record outage over a 24-hour period on Friday, with around 1.3m homes affected.
Portsmouth was relative unscathed when it came to power cut. But in Sussex, scores of homes in more rural areas remain in darkness.
Electricity provider Western Power Distribution (WPD) confirmed the outage was the most widespread ever recorded for the south west of England.
The company said: ‘Since it first hit, Storm Eunice has officially caused the highest number of power cuts in a 24 hour period our South West region has ever experienced.
‘Our engineers are continuing to work relentlessly to restore supplies to our customers despite the awful conditions.’
At the height of the storm, the roof of the O2 Arena in London was damaged – causing rapper Dave’s upcoming concerts to be postponed – and the spire of St Thomas Church in Wells, Somerset, crashed to the ground.
In Portsmouth, walls collapsed and trees were uprooted as winds approaching 70mph battered the island.
In Fratton Road, Fratton, a partially collapsed building remains cordoned off amid fears it could suffer a further collapse.
Met Office meteorologist Greg Dewhurst urged Britons to brace for more windy weather.
Speaking on Saturday, he added: ‘We will see a slight easing in the wind over the evening time tonight, but it’s not long before they pick up again tomorrow to lead to another windy day across the UK.
‘This will have an impact on the clearing up process over the course of the day.’
The Association of British Insurers indicated that the clean-up could cost more than £300m.
A spokesman said: ‘It is too early to estimate the likely insured cost of Storm Eunice, when insurers will be focusing on assessing damage and helping their customers recover.
‘No two storms are the same. The last significant storms to hit the UK – Ciara and Dennis – led to insurers paying out over £360m.’
National Rail has warned there is still ‘major disruption’ to train services ‘across most of Great Britain’.