The graph above shows the latest available data for Iraqi production. As you can see, there was a sharp increase in March and April. The April number is entirely dependent on one source - OPEC (secondary sources) - but post-invasion Iraq seems much more transparent than other middle eastern countries and the data sources don't vary that much. So I would expect this to be mostly confirmed.
According to Bloomberg, more is on the way:
The country is targeting production of 3.4 million a day this year and more than 4 million barrels in 2013, according to Asim Jihad, a spokesman for the Oil Ministry in Baghdad.and
Iraq, seeking to more than double oil output by 2015, is poised to overtake Iran as OPEC’s second- largest producer by the end of the year as sanctions hobble crude production in its Persian Gulf neighbor.My past coverage on prospects for Iraqi oil production can be found here.