He said the Fed has a list of job indicators, including long-term unemployment, it will be watching and the picture is still not strong enough to speed up the Fed's return to normalcy. "I think they need confirmation over the next three- to six-month period that we've jumped to this higher rate of job creation," Zandi said.
The Fed has been tapering back its bond-buying program in $10 billion increments, and it is expected to wind down the program in the fourth quarter. After that it would consider when to begin raising short-term interest rates.
"It's a long way before the Fed meets to raise interest rates. I think it will be by this time next year. That's the current forecast. We're not going to get jobs numbers like today in every number going forward," said Zandi.
Average hourly earnings rose by 0.2 percent, for a 2 percent increase year over year. The level of labor force participation, or share of working age Americans employed or looking forward, was unchanged at its post-recession low of 62.8 percent.
The number of long-term unemployed, out of work 27 weeks or longer, fell by 293,000 to 3.1 million. Fed Chair Janet Yellen has expressed concern about this group, which totals about a third of all unemployed and has the hardest time finding jobs.
"We're generating jobs by the bushel," said Ward McCarthy, financial economist at Jefferies. "Some of the things Janet Yellen is looking at are improving but they're not good enough. ... They're at distressed levels. They're still worse than we've seen in prior recessions."
"One fly in the ointment is the composition of the household survey. There was a big increase in part-time employment and decrease in full-time employment," he said.
Part-time workers increased by 275,000 in June to 7.5 million. The number of discouraged workers in June was 676,000, down 351,000 from the year earlier.
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—By CNBC's Patti Domm