According to Tolkien canon (e.g. sourced from JRRT ot CT), what was the earliest point that:
Saruman suspected that the One Ring was not fully lost and is recoverable
Realized that the One Ring was found (by Bilbo).
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According to Unfinished Tales
In the Tale of Years the entry for 2851 refers to the meeting of the White Council in that year, when Gandalf urged an attack on Dol Guldur but was overruled by Saruman; and a footnote to the entry reads: ‘It afterwards became clear that Saruman had then begun to desire to possess the One Ring himself, and hoped that it might reveal itself, seeking its master, if Sauron were let be for a time.’
The Fellowship of the Ring set out in 3018. Unfinished Tales also states that Saruman had already been searching the Anduin and had found the Elendilmir and a golden case where Isildur had carried the Ring. Of course the Ring itself had gone centuries earlier.