Freezing temperatures are due to continue throughout this week, with snowfall likely to hit the majority of the region throughout tomorrow.
Forecasters from the Met Office confirmed yesterday that up to six inches of snow is due to fall on higher ground in Yorkshire, with at least two inches expected across many parts of the region.
A senior Press officer for the Met Office, Barry Gromett, said: "While it might not be as bitterly cold as it has been, the freezing temperatures are expected to continue.
"We anticipate there will be heavy snowfall on Tuesday, and ice is due to cause further problems on the roads and railways."
Parts of the UK enjoyed a white Christmas, including coastal areas and higher parts of the Yorkshire region.
Edinburgh, Glasgow, Aberdeen and north-east England all claimed to have had some Christmas snow but forecasters said there was more snow on the ground than came from the air.
Most of the UK saw crisp wintry conditions with clear blue skies and freezing temperatures. It means this month is set to be the coldest December since records began.
While the coming days are expected to be slightly warmer than the bitterly cold snap that has gripped Britain, temperatures are still predicted to fall to -2C or -3C at night.
Senior fire officials have once again issued a warning about the dangers of venturing on to frozen lakes and rivers.
Crews in York had to mount two Christmas Day rescues, the first of which saw firefighters retrieve a dog that had walked on to a frozen stretch of the River Ouse near the city's Millennium Bridge at about 1.15pm. Only two hours later, fire crews rescued a 59-year-old man from the Ouse near Skeldergate Bridge at 3.15pm.