That's not the thing to do ! As you've discovered, the spider is not necessarily flushed away; there's generally a U-bend in the pipe, which saves it and it climbs back up; if you succeed, you've just killed one of our utterly harmless and very useful creatures, which is, understandably, a lot more frightened of you than you are of it.
Drape a towel over, near where it is, so it can climb up the towel and over the edge of the bath. You can always wash the towel afterwards, if you're that squeamish. It will then disappear. Remember that there are spiders in your home already and plenty of insects, which they live on, which you never see, except by chance. Set free, this one will go back to being out of sight.