Why did Sauron need ring?

[Sauron the White]

Question: much is made of the fact that if Sauron obtains the One Ring during the Lord of the Rings timeperiod, then all is lost and the world of Middle-earth comes under his shadow and rule. Here is what I do not quite get - Sauron had the One Ring. It was his in the Second Age and cut off of his finger by Isildur right there on his own doorstep in Mordor. So he had the ring, he had his armies, he had power. And it availed him not. The armies of the Last Alliance marched upon him, laid siege to his tower of Barad-dur and defeated him despite his possession of this powerful ring. So why then at the end of the Third Age does everyone with any knowledge seem to feel that the Ring will spell the end for the Free Peoples of Middle-earth when it had not had that effect previously?

There is talk of the power of Sauron having grown during the time - but how could Sauron without the ring have grown in more power compared to the time when he actually had the ring? That seems a huge contradiction.

Why did Sauron need the Ring at the end of the Third Age when he had it earlier and it failed to help him?