One group of 67 migrants, including 13 children, was found stranded in their wooden boat off the western Greek port of Methoni on the Peloponnese peninsula, facing southern Italy.
The migrants, whose nationalities have not been revealed, appear to have been trying to leave Greece for Italy, according to initial reports.
They were taken to Methoni to be processed and identified, according to harbour police, who revised their earlier figure of around 50 migrants rescued.
All 36 migrants on board, including seven children, were rescued by the local authorities.
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After a major lull since the European Union (EU) signed a controversial deal with Turkey to stem the migrant influx in March, there has been a slight uptick in arrivals in Greece in recent days.
Between Thursday and Friday morning, 261 new arrivals were registered on the Aegean islands -- mainly Lesbos -- an increase on recent days, according to the SOMP agency which is coordinating Greece's response to the migrant crisis.
Under the deal, Turkey agreed to take back Syrians who make it to Greece, in return for being allowed to send one from its massive refugee camps to the EU in a more orderly redistribution programme.
Lat week there were between 13 and 147 new arrivals every day.
Both Greece and the EU fear that the migrant floodgates could re-open as Turkey focuses on a purge of officials following the failed coup of July 15 which has led to a souring of relations between Ankara and Brussels.