If you can stand the hassle of ‘tyre kickers’ coming round to view and making silly low offers then, Autotrader is good. Autotrader is v good to help you get a good idea of your car’s value - use the many filters to find a car as near the exact spec, mileage and age of your vehicle.
Otherwise, part exchange is the far simpler option.
I use Ebay. The problem I find though is that people generally got ripped off when they bought the car so they have a completely unrealistic idea of it's value so tend to think that the offers they get are left field but actually are quite reasonable.
Among the advice on the AA website is "get a new MoT, especially if there's less than 3 months left". I thought the earliest you can get your car MoT'd is 4 weeks before its expiry?
^^^ You can get an MOT test done on your car every day of the year if you like, GG - or even several times a day if you're really odd!
However all that will happen when you do is that you'll keep getting fresh certificates that are only valid for exactly 12 months. If you wait until there's one month (less a day) before your existing MOT certificate expires, the unexpired time will be added to your new certificate, meaning that it will then be valid for 13 months.