The government report seeing ‘some recent misreporting we wish to clear up. You can take on other employment while on furlough, provided your existing employment contract permits it’
Some contracts may contain exclusivity clauses, however, if present they need to be reasonable. Any exclusivity clause In a zero hours contract are not valid since being banned by the government back in May 2015.
No valid exclusivity clause, then you are free to find alternate forms of income. Are you tied to hours availability – no as when on furlough you have no working hours with the exception of undertaking appropriate training.
Working for a different employer
For any employer that takes on a new employee, the new employer should ensure they complete the starter checklist form correctly. If the employee is furloughed from another employment, they should complete Statement C.
PAYadvice.UK 10/4/2020