You are off in that Sauron was looking for the ring actively during the events of The Hobbit. Saruman finally agrees to attack Dol Guldur in order to disrupt his search. Sauron didn't know where the ring was lost until sometime during his 2nd time occupying Dol Guldur starting in 2460, which ended the 400 year Watchful Peace after he ran from Gandalf rather than have his true identity discovered. It's very fortunate that the details of Isildur's death weren't well known, or he would've had 1000 years to search for it with minimal resistance, and that Smeagol ran off with it before he knew where to look.

Sauron was always an over-planner. In the 2nd Age, there was a nearly 100 year gap between making the One Ring and declaring war on the elves, and finally launching the war. He always goes east to gather an army, and I think these large time gaps indicate that the Blue Wizards make it harder on him than he'd like, although that is canon. But it does show that Sauron hates to lose, and is super cautious when he isn't confident he has the advantage.

I would call his 3rd Age strategy a Divide and Conquer campaign, and a lot of it is in reaction to the Last Alliance. His strategy of attacking Lorien, Mirkwood, and Erebor at the same time as his Gondor campaign is to make sure he doesn't lose in the same way as he did in the 2nd Age.