Do you always have to write a new SS for contests?

So far, yes.

But it depends what a new contest's rules stipulate. Usually they give you a guideline - 'Write this kind of short story' - and then you have to write whole new story like that.

However, there have been Epics contests where you simply entered an epic that you had already written and published here. It's possible there might be a short story contest like that one of these days.