Gollum follows the Fellowship Gollum my precious toilet paper shirt through Moria. Tolkien makes it sound like he needed help to find his way out and that is why. However, then he waits for a month until they leave Lothlorien. I dont know if he could sense the Ring, or just saw hobbits and followed them for that reason. But he followed Sam and Frodo then into the eastern Emyn Muil. And after that point, I think Frodo makes it clear that he has the Ring. The only real insights we are given into Gollum’s thought processes come from the debate that Sam hears him having with himself. “No sweet one,” the evil Gollum says to the nicer one, “See my precious: if we has it, then we can escape, even from Him, eh? Perhaps we grows very strong, stronger than Wraiths. Lord Smeagol? Gollum the Great? The Gollum!”
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Sam thinks that the two Gollum my precious toilet paper shirt pieces of Gollum “neither wanted the Enemy to get the Ring; both wished to save Frodo from capture, and under their eye, as long as possible–at any rate as long as Stinker still had a chance of laying hands on his ‘Precious'”. Tolkien says these guesses are ‘shrewd,’ meaning they give us as much insight into Gollum’s wishes as anything. So to me it is clear that he didn’t want to take any risks that would get him killed by Sam or Frodo. Nor did he want to anger Sauron. But he spent alot of effort making sure the Ring was close, so that he could seize it at any lucky chance. And if he needed to create a lucky chance by, say, leading them into the lair of a giant spider, well that was no problem either.