Mr Stephenson headed the council nearly two decades ago and came back to run the authority last summer following the sudden departure of chief executive Nicola Yates from her £160,000-a-year job.
The council advertised for a “dynamic” new chief executive in the summer but failed to recruit.
Mr Stephenson, who has been working the equivalent of 22.2 hours a week, was paid a salary package of £40,671 in the financial year 2012-2013.
The authority is also looking to appoint new chief operating officer internally on a salary of around £130,000 who would be expected to take overall responsibility for the day-to-day running of the council when Mr Stephenson wasn’t there.
The Liberal Democrats said they were concerned that the two roles in combination could end up costing the taxpayer more than Nicola Yates, who was on £160,000.
Deputy group leader Coun Mike Ross said: “We have been clear that this shouldn’t end up costing more than one chief executive and we seem to be left with a situation that’s going to end up costing the taxpayers more. Labour has to take the blame for that.”
However council leader Steve Brady insisted that the total bill, including the part-time chief executive, would be lower than any other chief executive in the country. He said a suggestion that the new chief operating officer would be on £130,000 was “pure conjecture.”
Coun Brady said the post would be for two years, adding: “Darryl has accepted the post. He will be on two days a week. There are no pension costs whatsoever. If you look at the East Riding the total (chief executive’s) package has been reported as £202,000.
“He has been absolutely superb over the last 18 months. He is recognised as an excellent chief executive and I am pleased he is stopping on for the next two years. What I need as a leader is someone I have total confidence in - and I have that in Darryl Stephenson.”