Almost two-thirds of young people under the age of 25, 58.8 percent, were without a job.
The new figure was higher than the 27.6 percent rate recorded in May and the 24.6 percent the previous year.
The number of jobless stood at just over 1.4 million, an increase of 174,709 persons compared with June 2012, the data showed.
Highly indebted Greece has been implementing unpopular austerity measures demanded by the European Union and International Monetary Fund in exchange for a multibillion-euro bailout. It is in its sixth year of recession.
The conservative-led government has promised to end the recession next year.
NJF/NJF