The search for the one ring

Was anyone looking for the one ring to destroy before Gollum or Bilbo found it?

I know Sauron wasn't particularly prevalent after Isildur took the ring, but I was just curious as to whether anyone was looking for it after he died, you know, just to make sure Sauron never came back?

In the Hobbit, Saruman says that the enemy has been vanquished, so was it believed that Sauron was gone for good?