[Mithadan]

Just speculating, but Gandalf had other reasons to visit Gondor and review its archives other than seeking information about the Ring. Gandalf knew that Sauron had arisen and occupied Dol Guldor well before Bilbo found the Ring. Was he interested in the Ring? Certainly. But he had been assured that the Ring would not be found by Saruman. Nonetheless, Sauron was back, and a war seemed inevitable even without the Ring. Certainly, Gandalf would want to assess Gondor, the first line of defense, and acquire any information that might be helpful in dealing with Denethor. This may have included history and legends. In addition, Aragorn (II) was born in TA 2931 and Gandalf would have been interested in the dealings between the Stewards and Isildur's heirs over time.

There were a number of reasons for Gandalf to visit Gondor after Sauron arose, and any combination of them would be a good reason for his research. Add to this Gandalf's likely growing suspicion of Saruman. He may have wanted to double-check what Saruman had represented.