From the 1st July, you can start to bring back furloughed staff part time and use the Flexible Furlough Scheme to cover some of their wages.
There is no need for a new contract between you and your employees, although the guidance says that the working hours have to be set out, at least every week but this could be in a letter or email.
Employment law experts Direct Law and Personnel, is providing complimentary resources to help employers during the coronavirus outbreak. A Flexible Furlough Letter can be downloaded HERE, this will be needed for the employee and employer to be on the same page but also as evidence for HMRC if it is ever needed.