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.
The migrants faced down pepper spray used by police as they broke out of a holding centre in a cornfield and marched down a motorway towards Budapest. They later agreed to be taken by bus to another reception centre.
Further south, the BBC's James Reynolds reports long queues of migrants waiting to pass from Macedonia into Serbia on the border town of Presevo.
Hungary is a bottleneck 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 surprised even his own colleagues when he resigned on Monday, in a move correspondents say was clearly related to problems with the construction of a border fence meant to keep migrants out but which has so far proved ineffective.
Attempts to block the passage of migrants in Budapest failed on Friday, and some 20,000 migrants made their way from Hungary into Austria and Germany over the weekend.
German Chancellor Angela Merkel has warned that the "breathtaking" flow of migrants into Germany will change the country in the coming years.
On Monday, officials said that the processing centre on Lesbos would operate around the clock for five days.
But on Monday night, about a dozen coastguards and riot police armed with batons struggled to control some 2,500 migrants surging towards one such ship, reported AFP news agency.
Local authorities have been overwhelmed by the migrants - mainly from Syria, say officials - 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".
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 urged other EU states to help shoulder the burden amid opposition criticism of its open-border policy.
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.
However, Hungary, along with the Czech Republic, Slovakia and Romania, has rejected the idea of official quotas.
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