Around 14,000 people crossed the Evros frontier from January through September of this year, more than double the number for the whole of last year, according to the Greek police. Around half of them were Turkish citizens, according to estimates from Frontex, the European Union’s border agency. Many are judges, military personnel, civil servants or business people who have fallen under Turkish authorities’ suspicion, had their passports canceled and chosen an illegal route out.

Nearly 4,000 Turks have applied for asylum in Greece so far this year. But most Turkish arrivals don’t register their presence in Greece, planning instead to head deeper into Europe and further from Turkey.

“We didn’t say goodbye to anyone before leaving,” said Mr. Cagar, who is now in Athens trying to find some way to get to Germany. His wife and child already made it there with the help of smugglers who have demanded a hefty price. “We began our journey with €13,000 ($14,700) and I have €1,500 left,” he said.

Ahmed, a 30-year-old former F-16 pilot in the Turkish air force, spends his days talking to smugglers and trying to find a way out. “My dream is Canada, but the reality is Omonoia,” he said, referring to the gritty square in downtown Athens where migrants and smugglers mingle.

A few months after the coup attempt, Ahmed said, he was dismissed, accused of Gulenist links, arrested and beaten, after another officer denounced him. He said he has no connections with Mr. Gulen’s network. He was released pending trial, but decided to flee when a prison term appeared unavoidable.

Yilmaz Bilir, his wife Ozlem and their four children were on vacation when the coup attempt happened. Mr. Bilir, who worked at the information-technology department of Turkey’s foreign ministry, found out months later that he was suspected of Gulenist links, which he denies. The family went into hiding, staying with relatives and friends. Mr. Bilir was arrested when he briefly visited his own home and neighbors called the police. When he was released pending trial, the family decided to leave Turkey.

Mr. Bilir made it to Germany using a forged passport and has applied for asylum there. His wife and children have applied to join him.

Mrs. Bilir, stuck for now in Athens, remembers how happy the family was when they crossed the river Evros one summer night. “It was an endless walk, but we were happy, because we were away together,” she said. “I was so stressed in Turkey that I couldn’t sleep well for months, but that first night in detention in Greece, I finally slept.”

After the coup, Meral Budak was suspended from her job as a teacher. Her husband was a journalist at Zaman, a major Turkish newspaper linked to Mr. Gulen’s movement. He had a valid U.S. visa and was able to travel to Canada, where he now works as an Uber driver. His 18-year-old son joined him a few months later.

Mrs. Budak and the couple’s 15-year-old son Ali remained in Turkey and soon had their passports revoked. They went into hiding for a year. “The most traumatic memory was when I burned hundreds of books,” she said. “Even my children’s school books could be considered evidence, since the publishing companies were funded by Gulen.”

“When I was walking through Greek villages, I realized my life was never going to be the same,” Mrs. Budak said. “I was walking into the unknown.”