Officials from Downing Street and the Department of Trade and Industry contacted both sides in the bitter dispute ahead of a third day of talks.
Some 500 workers staged a lightning protest in Coventry followed by action in Warrington; there are an estimated 20,000 on strike in London and south-east England.
With deliveries to millions of homes halted by the unofficial dispute, Ms Hewitt made it clear that she wanted the dispute resolved.
A spokesman for the minister, who is sole shareholder in the public corporation on the government's behalf, said Ms Hewitt believed it was "important to resolve this as quickly and amiably as possible".
The action started nearly a fortnight ago when a driver in the capital, returning from an official strike over London weighting, was suspended.
Tens of millions of letters remain undelivered in much of London and areas including Southend, Chelmsford, Dartford, Oxford and Milton Keynes.
About 30 drivers walked out in Portsmouth yesterday as did colleagues in Langley, Berkshire, and Leigh-on-Sea, Essex. Employees in parts of Scotland briefly walked out earlier this week before returning to work.
Special deliveries, whereby customers pay extra to guarantee the arrival of letters and packets within a certain timeframe, have been suspended in London and post boxes are being sealed.
Malcolm Bruce, the Liberal Democrat trade spokesman, said: "A national universal postal service is part of Britain's critical infrastructure. As a publicly owned public service, the government has a responsibility to help bring this current dispute to an end.
"If compulsory arbitration is required to tame the wildcat strikes, then Patricia Hewitt must be prepared to impose it."
David Frost, director general of the British Chambers of Commerce, said: "This issue goes beyond jobs in the Royal Mail and is affecting jobs across the whole economy. Businesses are suffering as a result of the wildcat strikes."
