[Gordis]

I have once participated in a similar discussion and I have a list of quotes from the Tale of Years, the main text of LOTR and The Hunt for the Ring in UT concerning Gollum. I will post it here, perhaps it will be of use.

Tale of Years:
2941 Bilbo meets Sméagol-Gollum and finds the Ring

2944 Gollum leaves the Mountains and begins his search for the 'thief' of the Ring.
[LOTR: "He found his way into Mirkwood;… his padding feet had taken him at last to Esgaroth, and even to the streets of Dale, listening secretly and peering. Well, the news of the great events went far and wide in Wilderland, and many had heard Bilbo’s name and knew where he came from. We had made no secret of our return journey to his home in the West. Gollum’s sharp ears would soon learn what he wanted.’
‘Then why didn’t he track Bilbo further?’ asked Frodo. ‘Why didn’t he come to the Shire?’
‘Ah,’ said Gandalf, ‘now we come to it. I think Gollum tried to. He set out and came back westward, as far as the Great River. But then he turned aside. He was not daunted by the distance, I am sure. No, something else drew him away. So my friends think, those that hunted him for me.
‘The Wood-elves tracked him first, an easy task for them, for his trail was still fresh then. Through Mirkwood and back again it led them, though they never caught him. …But at the western edge of Mirkwood the trail turned away. It wandered off southwards and passed out of the Wood-elves’ ken, and was lost. … But I am afraid there is no possible doubt: he had made his slow, sneaking way, step by step, mile by mile, south, down at last to the Land of Mordor…. Mordor draws all wicked things, and the Dark Power was bending all its will to gather them there. The Ring of the Enemy would leave its mark, too, leave him open to the summons. And all folk were whispering then of the new Shadow in the South, and its hatred of the West. There were his fine new friends, who would help him in his revenge!]

2951 Sauron declares himself openly and gathers power in Mordor He begins the rebuilding of Barad-dûr and sends three of the Nazgûl to reoccupy Dol Guldur. Gollum turns towards Mordor.

2980 About this time Gollum reaches the confines of Mordor and becomes acquainted with Shelob.
[LOTR: Already, years before, Gollum had beheld her, Sméagol who pried into all dark holes, and in past days he had bowed and worshipped her, and the darkness of her evil will walked through all the ways of his weariness beside him, cutting him off from light and from regret.] .
[(LOTR: Orc talk)..But Shelob was on the go. My lads saw her and her Sneak.'
`Her Sneak? What's that? '
`You must have seen him: little thin black fellow; like a spider himself, or perhaps more like a starved frog. He's been here before. Came out of Lugbúrz the first time, years ago, and we had word from High Up to let him pass. He's been up the Stairs once or twice since then, but we've left him alone: seems to have some understanding with Her Ladyship. I suppose he's no good to eat: she wouldn't worry about words from High Up.]

3001 Gandalf seeks for news of Gollum and calls on the help of Aragorn.

3009 Gandalf and Aragorn renew their hunt for Gollum at intervals during the next eight years, searching in the vales of Anduin, Mirkwood, and Rhovanion to the confines of Mordor.
At some time during these years Gollum was captured by Sauron. Gollum must have been all around the confines of Mordor, for 37 years, first close to Cirith Ungol worshipping Shelob, then he seems to wander to Morannon (Why and what for?). It looks like Gollum was captured on the road east of Morannon: [LOTR: "Back a little, and round a little' – his skinny arm waved north and east – `and you can come on hard cold roads to the very gates of His country... His Eye watches that way all the time. It caught Sméagol there, long ago."
[UT: Gollum was captured in Mordor in the year 3017 and taken to Barad-dûr, and there questioned and tormented. When he had learned what he could from him, Sauron released him and sent him forth again. He did not trust Gollum, for he divined something indomitable in him, which could not be overcome, even by the Shadow of Fear, except by destroying him. But Sauron perceived the depth of Gollum's malice towards those that had "robbed" him, and guessing that he would go in search of them to avenge himself, Sauron hoped that his spies would thus be led to the Ring.]

3017 Gollum is released from Mordor. He is taken by Aragorn in the Dead Marshes, and brought to Thranduil in Mirkwood. (Gandalf questions him there). Gandalf visits Minas Tirith and reads the scroll of Isildur.
[(LOTR: Gollum) 'A little path leading up into the mountains: and then a stair, a narrow stair, O yes, very long and narrow. And then more stairs. And then' – his voice sank even lower – `a tunnel, a dark tunnel; and at last a little cleft, and a path high above the main pass. It was that way that Sméagol got out of the darkness]
UT: [After his release from Mordor] Gollum soon disappeared into the Dead Marshes, where Sauron's emissaries could not or would not follow him.

20.06.3018 Sauron attacks Osgiliath (And the nazgul set out for the hunt for the Ring-UT). About the same time Thranduil is attacked, and Gollum escapes.

August 3018 All trace of Gollum is lost. It is thought that at about this time, being hunted both by the Elves and Sauron's servants, he took refuge in Moria; but when he had at last discovered the way to the West-gate he could not get out.