Some 25,717 individual insolvencies were recorded, showing a 3% increase on the previous three months, although this is still 6% lower than the same period last year, the Insolvency Service's report said.
The overall figure masks a decline in bankruptcies, which fell by 3% on the previous quarter and remained at their lowest levels for more than a decade.
Some 6,469 bankruptcy orders were recorded in the second quarter of this year, putting them one fifth lower than a year ago.
Bankruptcies are often seen as a "last resort" and their use has generally dropped off since the introduction of a newer type of insolvency called a debt relief order (DRO), which is aimed at people with lower levels of debt but no realistic prospect of paying it off.
Around one quarter of bankruptcy orders involve people who are self-employed, which is a bigger proportion compared with previous years due to a decline in cases involving people who are not traders, the Insolvency Service said.
The recent upturn in personal insolvencies was driven by a rise in i ndividual voluntary arrangements (IVAs), which are agreements for people to pay back as much as they can into a pot which is shared out between creditors.
DROs, which are often dubbed "bankruptcy light", fell back to their lowest levels in more than two years. Their numbers dropped by 1% on the previous quarter, with 7,132 recorded.