Authorities reopened some airspace and airports after emergency talks between European governments, airlines and regulators resulted in a plan that divided the continent into safe and unsafe areas.
Dublin airport was closed from 1800 GMT on Sunday on Monday due to the ash cloud, after airports in the northwest of the country had been closed earlier in the day.
"The outlook [for] later tomorrow looks better, I wouldn't be optimistic for the early part of the day but the later part of the day looks better and as the week goes on, it should improve," Eamon Brennan, the the head of the Irish Aviation Authority (IAA), told national broadcaster RTE.
It was later announced that Dublin would remain closed until 1100 GMT on Wednesday.
Shannon, an important possible stopping point for flights to the United States, would be open until 2200 GMT, the Irish Aviation Authority said.
"North Atlantic overflights through Irish-controlled airspace remain unaffected," the IAA said in a statement.
The UK government said on Saturday warned that parts of British airspace might have to close at various points until Tuesday with different parts likely to be closed at different times.
Britain's Met Office, which monitors weather conditions, said: "Winds are expected to blow mainly from the northwest for a time over the weekend with the risk of ash affecting some parts of the UK.
"However, winds are predicted to swing into a south westerly direction by the middle of next week, which would take most of any ash away from the British Isles."
The volcano under the Eyjafjallajokull glacier in Iceland is continuing to erupt with no signs of that the activity is about to end, but Agust Gunnar Gylfason, an local civil protection office, said that there had been little change in intensity.
"What really changes the situation is the weather pattern," he told The Associated Press news agency.
Closures 'a joke'
Richard Branson, the president of Virgin Atlantic, and British Airways have both criticised the latest restrictions.
"The closing of Manchester airspace once again is beyond a joke," Branson said in a statement. He said test flights have "shown no evidence that airlines could not continue to fly completely safely".
British Airways said that the approach was "overly restrictive and not justified on safety grounds".
Jonathan Nicholson, a spokesman for Britain's civil aviation authority, called Branson's remarks "surprising" after last week airline representatives and engine manufacturers had agreed to find a way to ensure planes could fly safely in the volcanic ash.
"We as an organisation can't just say, 'Oh, I'm sure it's all right, go fly without evidence it's safe,'" he said.
Iata, the international airline industry body, said last month's shutdown had cost airlines about $1.7bn and called on governments to pick up at least part of the tab.
Eurocontrol said more than 100,000 flights to, from and within Europe had been cancelled between April 15 and 21, preventing an estimated 10 million passengers from travelling.