The Greek government and the UN refugee agency have brought in extra staff and ships to deal with some 25,000 stranded migrants on the island of Lesbos.
A processing centre has been also set up on an abandoned football ground to help the migrants to get to Athens.
A Greek minister said on Monday Lesbos was "on the verge of an explosion".
Meanwhile, hundreds of migrants broke through police lines on Hungary's border with Serbia and started walking towards the capital, Budapest.
Scuffles erupted at the holding centre in Roszke as migrants tried to force their way past police. Stones were thrown at officers, who responded with pepper spray.
About 300 were later seen walking along a motorway, escorted by police officers.
However, as darkness fell they reportedly agreed to be taken by bus to a reception centre.
Hungary has become a flashpoint as thousands of migrants and refugees from the Middle East and Africa move north to claim asylum in Germany and other countries.
Hungarian Defence Minister Csaba Hende resigned on Monday, reportedly over problems with the construction of a border fence meant to keep migrants out.
Hungary had previously blocked those heading north, insisting they be registered there first as required under EU rules. But it dropped restrictions on Friday after struggling to cope with thousands camping in Budapest.
About 20,000 migrants made their way from Hungary into Austria and Germany over the weekend.
Separately, German Chancellor Angela Merkel warned that the "breathtaking" flow of migrants into Germany would change the country in the coming years.
In other developments:
--The UK will accept up to 20,000 refugees from Syria over the next five years,David Cameron has told MPs
--At least 150 migrants in southern Denmark have tried to march towards the border with Sweden, forcing police to close a motorway
--The US administration "is actively considering a range of approaches to be more responsive to the global refugee crisis, including with regard to refugee resettlement", a White House spokesman has said
On Monday, officials said that the processing centre on Lesbos would operate around the clock for five days.
The hope is that it will help people to buy tickets for specially chartered ships to get to Athens, the BBC's Jonny Dymond on Lesbos reports.
Local authorities have been overwhelmed by the migrants who have been forced to live in squalid conditions, our correspondent adds.
Greek Migration Minister Yiannis Mouzalas warned that the island was "on the verge of explosion".
Most of migrants are from the war-torn Syria, officials say.
An estimated 340,000 asylum seekers have arrived in Europe so far this year, most braving dangerous sea journeys from North Africa and Turkey.
Germany, where most migrants are headed and which expects 800,000 asylum requests this year, has said it wants other EU states to help shoulder the burden. But the crisis has divided the 28-nation bloc.
French President Francois Hollande said mandatory quotas were being drawn up to relocate 120,000 migrants across the EU, and that France would take 24,000.
Earlier, s Merkel thanked volunteers who had welcomed those arriving over the weekend, saying they had "painted a picture of Germany which can make us proud of our country".
However, she said that although Germany was "a country willing to take people in", it was "time for the European Union to pull its weight".
Merkel is facing criticism at home over Germany's willingness to accept so many asylum seekers.
Hungary, along with the Czech Republic, Slovakia and Romania, has rejected the idea of official quotas.