Airlines in northern Europe have started flying again, with air traffic agency Eurocontrol saying it expects up to 60% of daily flights over Europe to go ahead.
But many planes remain grounded and London's airports are still closed because of a new ash cloud spreading south and east from Iceland.
Qantas says a new partial exclusion zone has been established but the airline is seeking more information on how it will affect travel.
Air New Zealand says passengers should not start their travel if their final destination is Britain or Europe.
The European Union says there were shortcomings in the way parts of Europe's air space was closed six days ago.
EU Transport Commission official Helen Kearns says the decision-making process has been sufficiently co-ordinated.
The restrictions were imposed six days ago because of fears ash from the volcano could damage aircraft engines.
EU transport ministers said there would be a core no-fly zone, a "caution" zone allowing some flights and another zone open to all flights, the BBC reports.
However, Britain later scaled back plans to reopen its airspace as the eruption strengthened, sending more ash towards Britain.
The National Air Traffic Services, which manages British airspace, said the latest Met Office information showed the situation worsening in some areas.
Airspace in Scotland and parts of the north of England was still expected to reopen on Tuesday, but plans for flights to resume at airports in Northern Ireland and as far south as London looked doubtful.
Spain has 17 airports open and has suggested its airports could be used an an entry point to Europe. Romania has also opened its airspace.
A United States official said on Monday that a Nato F-16 fighter jet had suffered engine damage after flying through the volcanic ash cloud.
The International Airways Volcano Watch group - a division of the International Civil Aviation Authority - says it has long warned of the need to prepare for a major volcanic event.
Airline costs mount
The International Air Transport Association on Monday estimated airline revenue losses were reaching $US250 million a day, up from an earlier estimate on Friday of $US200 million.
Australian carrier Qantas says it is losing about $1.9 million a day, with additional losses from air freight contracts which can't be met, while Emirates has a fifth of its fleet grounded, and believes its losses are $70 million a day.
Air New Zealand has revealed in its monthly investor update that the chaos is costing it $500,000 a day.
