If there is a good reason why you have to leave your current accommodation (e.g. end of lease), you can make a application for Rent Supplement in respect of a new address, once it's still within the max rent limits (note that new limits apply from Jan 2012).
You will have to go through the application process again, getting application forms for yourself and your new landlord. There may be a gap in payment while your new application is processed - be sure you sort this out with your landlord in advance.
If you qualified for Rent Supplement based on housing need (i.e. you are on a local authority housing list) and you are moving to a new local authority area, you will have to make a fresh application with the new local authority.