Most important thing is that you must be truthful.
If you are applying in Sep, and starting the job in Oct - the max you can write is that you are incoming into that company. That too when you have the offer contract. You would not be able to write anything related to your role or duties since you haven't started. Also, the decision of whether to mention the job at all will depend on the job itself. If it is going to add value to your application then sure why not? If you are joining a job completely unrelated and far away from consulting, and then applying to consulting - thats not going to be much of a value-add.
If you are applying in Oct, and starting the job in Oct - you could mention some of your roles and responsibilities. However, be mindful that you should not write a truck-load of stuff since you have just started working. Be realistic and honest about what you write. Eventually everything on your CV must be defendable.