The downbeat showing comes just days after European Central Bank President Mario Draghi signalled one of the biggest policy reversals of his eight-year tenure and said the Bank would ease policy again if inflation failed to accelerate.
Despite years of ultra-loose monetary policy, prices have failed to rise as fast as the ECB wants, while a slew of recent data have suggested growth is slowing.
IHS Markit's Flash Composite Purchasing Managers' Index (PMI), which is considered a good guide to economic health, only nudged up to 52.1 this month from a final May reading of 51.8, beating the median expectation in a Reuters poll for 51.8 and its highest since November.
"It is nice it's moving in the right direction but you are only looking at GDP growth just over the 0.2% level. And that growth is unbalanced -- manufacturing is still in a downturn and we are reliant on the service sector," said Chris Williamson, chief business economist at IHS Markit.
Williamson's forecast is below the 0.3% predicted in a Reuters poll last month.
Firms in the bloc's dominant service industry staged a modest upturn, with that PMI rising to 53.4 from May's 52.9, ahead of expectations for no change.
But manufacturing activity contracted for a fifth month. The factory PMI held well below the 50 mark separating growth from contraction, registering 47.8 compared to last month's 47.7 and missing expectations for 48.0.
An index measuring output, which feeds into the composite PMI, dipped to 48.8 from 48.9.
Demand declined for a ninth month and as they have since September, factories were running down backlogs of work to stay active. The new orders sub-index held steady at May's 46.6.
As demand also remained weak for services, firms there barely increased headcount this month. The employment index fell to 53.8 from 54.0.
With forward-looking indicators painting a downbeat picture, optimism waned. The composite future output PMI fell to 58.7 from 59.8, its lowest reading since October 2014.
"There are a lot of concerns -- the weak state of the economy, both domestic and global; trade wars; weaker global demand conditions as well as rising geopolitical risks," Williamson said.