The refusal document says on the back 'if you intend to use your vehicle on the road,you should have it repaied without delay and retested before the existing test validity expires'
That implies to me that the previous MOT IS still valid at the time you are reading the refusal and you can still drive it. So you can get the test done 1 month early and have a month to fix any issues - such as the op's exhaust.
Also,
The MOT web site says
It is generally an offence to use on a public road, a vehicle of testable age that doesn’t have a current test certificate, except when:
-taking it to a test station for a test booked in advance
-bringing it away from a test station after it has failed the test, to a place of repair
-taking it to or bringing it away from a place where, by previous arrangement, repairs are to be made or have been made to fix the problems that caused the vehicle to fail its test
the last bullet says that you can take it to be repaired.