"They usually try to hide on trucks, but we now also find them stacked in car boots, risking their lives in doing so," said Ioakim Tsouknadis, deputy chief of police at Alexandroupolis.
And increasingly it is the immigrants themselves who organize the crossings, with Iraqis now competing with Bulgarians, Turks and even Greeks on their home turf.
"They end up knowing the local mountains like the back of their hand," said Tomazis.
The Filakio facility is Greece's newest, and is already full. It opened only a few weeks ago, amid complaints from human rights groups about conditions at other centers.
Last month, European Parliament deputies who inspected a detention center on the island of Samos reported "horrid" overcrowding and poor access to medical care and legal assistance.
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