The jobless total is now 1.85 million, after the second rise in two months, the first time there have been two consecutive jumps for two years.
The Office for National Statistics (ONS) said it was possible the jobs market was "levelling off".
But Paul Archer from Christians Against Poverty's Jobs Clubs told Premier's News Hour people shouldn't panic.
He said: "I don't think it's all doom and gloom, it's great to acknowledge the amazing increasing in employment we've had over the last year in total, unemployment has gone down by 221,000.
"I do think the economy is still looking strong, productivity seems to still be growing, it might have slowed slightly but there are a lot of vacancies still available, almost near record high level of vacancies at the moment."
Business groups said the figures were disappointed, while a union leader warned of the "first crack in the edifice of post-election Tory economic triumphalism".
The Government pointed out there were nearly two million more people in work than in 2010, with almost half a million more jobs in the private sector over the last year.
Figures from the Office for National Statistics showed that the number of people claiming jobseeker's allowance fell by 4,900 last month to 792,400 following a slight increase in June.
There were 31 million people in work in the latest quarter, 63,000 fewer than the three months to March, but 354,000 more than a year ago.
A record 14.5 million women are in work, while employment for men slumped by 71,000 to just under 16.5 million compared with the previous quarter.
ONS statistician David Freeman said: "This is now the second consecutive time we've reported fewer people in work on the quarter.
"While it's too early to conclude that the jobs market is levelling off, these figures certainly strengthen that possibility. Growth in pay, however, remains solid."