Off the top of my head, the *purpose* of the journey must be to get it MoT'd (and co-incidental other purposes do not negate this - such as a minor detour to buy milk from the supermarket on the way home). What is the [main] purpose, and what is co-incidental is a matter of fact and degree.
The purpose of the exemption is to enable vehicles without a current MoT to be MoT'd, but off the top of our collective heads, there is no strict requirement that the current MoT must have expired.
There might be a legitimate reason why the OP wants to MoT it before the current MoT runs out - perhaps the test will be within a month of the expiration of the current one, or he is trying to sell it with a full 12 months MoT, but in my experience, people with a legitimate purpose in mind, tend to share it as relevant background information, when asking the question.
However, if the OP has some other [main] purpose in mind, and is hoping that pre-booking an MoT will give him a free pass to use the car for that purpose, off the top of my head the exemption would not apply, even if he tricked us into telling him that it would.
