BBC weather forecaster Tomasz Schafernaker said the entire UK was likely to witness heavy frosts on Tuesday night, and there was a possibility of more significant, heavy snow showers in England and Wales at the end of the week.
"There is a risk of round two," he said.
More than 6,000 UK schools were shut on Tuesday, including 578 in Wales, 450 in Hampshire, 428 in Birmingham, 400 in Essex and 350 in Leicester and Rutland.
Authorities including East Sussex, West Sussex, Somerset, East Staffordshire, Dudley, and Kent also closed schools.
Authorities have already announced that nearly 40 schools in Wales will remain closed on Wednesday.
Margaret Morrissey, of the Parents Outloud campaign group, said the decision to keep thousands of schools shut for a second day sent the wrong signals to children.
She added: "We are giving children the message that when things get difficult, you should just stay at home and have fun."
Schools Secretary Ed Balls said he understood the frustration of working parents but believed many would have welcomed an early decision on school closures:
"Many parents, wanting to plan their arrangements for today, will be pleased to have certainty yesterday of what was going to happen," he said.
In other developments:
• Two men have been airlifted to hospital following two separate incidents in which they have fallen from toboggans, in Sedgley, West Midlands. They both suffered back injuries.
• Two military rescue helicopters assisted ambulance services in parts of Devon and Cornwall as roads became impassable following 10cm of snow in some areas.
Meanwhile, trains on the East Coast Main Line were disrupted, with at least nine rail services between London and Leeds cancelled.
Services were badly affected on Southeastern and South West Trains into and out of London, with suspensions or reduced timetables on most routes.
Southern has now resumed operation of some services from Kent and the south coast, although suspensions are still in place on the London to Brighton route and others around Greater London.
The Association of Train Operating Companies (Atoc) said all operators should be running a normal, full-time service by Wednesday.
Eurostar services from St Pancras International are still suffering disruption and all train passengers are advised to check with National Rail Enquiries before setting out.
Transport for London said most bus routes were operating but three remained suspended, while all London Underground lines were now running, although experiencing delays.
Heathrow, Stansted, Gatwick and London City airports are all open but have suffered a second day of "significant delays and cancellations".
Flights at Bristol, Glasgow, Edinburgh, Southampton and Aberdeen have also suffered disruption.
British Airways said diversions on Monday meant many planes had not been in the right places for their departures on Tuesday.
The Highways Agency is continuing to warn drivers in the worst-affected areas to travel only if their journey is essential.
Those who do venture out should be well prepared, it said.
We've got enough grit to make sure that we can grit continuously for the next 12 days
Jenny Hart, Highways Agency
The number of car journeys made across its network on Monday evening was 31% down on normal levels, according to the agency. Tuesday's rush hour was 26.5% down on a normal morning.
An agency spokeswoman said 500 gritting lorries were being used to keep main routes open.
Jenny Hart said: "We're well prepared for everything that the snow throws at us at the moment. We've got enough grit to make sure that we can grit continuously for the next 12 days - that's 18,000 tonnes of grit just in Yorkshire alone."
All the main roads remain open in Scotland, which avoided the severe weather seen in England and Wales and saw temperatures mostly stay above freezing overnight.
The extent of the disruption to the transport network has prompted some criticism of the UK's preparations for the snow.
BBC transport correspondent Tom Symonds said it was clear that local authorities did not have the resources to cope - but he said many experts believed it would not be worth the extra investment given the infrequency of such conditions.
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