Oh, I see. Other websites give you the 'password' to this website.
But only if neither site uses salt, so salt all your hashes. (Not even a salt is necessary here, just a site-wide addition would be fine. Or honestly you could just use a hash that wouldn't be used by a site too dumb to salt their database.)
There is zero downside to doing an extra hash, except the chance that someone codes such a basic thing wrong.