About 3,000 properties are without water because of burst pipes as snow and ice begins to melt.
Welsh Water is dealing with 200 incidents and warned of further possible problems as temperatures rise.
Gwynedd, Carmarthenshire, Ceredigion and Pembrokeshire are still affected, with people in the latter county left waiting for a repeatedly-delayed delivery of bottled water.
Most schools have reopened, but snow and ice meant more than 100 were shut.
Welsh Water said people may see discoloured water as supplies came back and it hoped to have everyone reconnected by Tuesday afternoon.
However, people in some parts of Pembrokeshire will not be able to get bottled water delivered until at least Tuesday.
It was originally planned that a Welsh Water delivery for Simpson Cross, St Davids, Roch, Croesgoch, Trefin and Porthgain would happen on Monday evening.
But Pembrokeshire council said the delivery would not arrive at its depot until midnight, meaning the water would not be dropped of until 08:00.
Meanwhile, council leaders are seeking extra cash from the Welsh Government to help them cope with the cost of dealing with the snow disruption.
Debbie Wilcox, Welsh Local Government Association leader, said councils were "faced with significant additional costs incurred over the last few days and in coming weeks which will be an extra pressure on already tight budgets".
Welsh Water said it was prioritising vulnerable people who were without water and opening bottled water stations to help those who were cut off.
Managing director Peter Perry told BBC Radio Wales' Good Morning Wales programme: "We'd normally see around 75 bursts or leaks on our system every day. We're currently dealing with 200."
He said problems were caused by older, cast iron pipes being damaged by ground movement caused by the cold weather.
Mr Perry urged customers not to leave taps running unnecessarily because supplies were low due to a surge in demand in the past few days.
Dee Valley Water also called on customers to report leaks caused by frozen pipes thawing so it could locate them quickly and repair them.
Darrin Pascoe, from Blaenwaun, near Whitland, Carmarthenshire, said he had been without running water since Thursday and had been filling up bottles from a neighbour's tap.
"I cannot wash or wash clothes or dishes. I cannot clean the kitchen," he said.
Meanwhile, Ian Cole, 68, said his home in Llanstephan near Builth Wells, Powys, was still cut off on Monday, due to "shoulder-height" snow drifts.
He claimed Powys council is yet to grit the area to allow cars to pass and he was running out of medication.
"In some places the drifts are shoulder-height, we personally are cut off," he said.
Most schools reopened, although more than 114 remained closed on Monday, mainly in Caerphilly, Blaenau Gwent and Torfaen.
The majority plan to reopen on Tuesday, with just tens of schools to stay closed, mostly in Blaenau Gwent.
Arriva Trains Wales said it planned to run a full service on the majority of its network on Monday, although no trains will run on the Heart of Wales or Blaenau Ffestiniog lines. It said the Heart of Wales line would stay closed on Tuesday.
"We would like to thank our customers for their continued patience over the past four days of disruption and our railway teams, who have been working tirelessly in very challenging weather conditions," a spokesman said.
In Tywyn, Gwynedd, councillors estimate as many as a thousand trees were felled because of the storm, with some damaging gas lines and school property.
Cardiff Airport said some flights may still be disrupted due to the knock-on effect of delays and cancellations in the past few days.
Meanwhile, organisers of the Newport Half Marathon, which was cancelled on Sunday due to the snow, confirmed it has been rescheduled for 18 March.
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