Yes. They usually send you a reminder letter just a few days before your sick note expires (which isn't much use), so it's best to send in a new one before the old note expires.
Hi Pusscat, it's worth asking your doctor for a new fit note but just to be aware, they don't like writing them before the current one is just about to run out.
Hopefully your doctor wont be as fussy as the ones I have seen!