Gollum exiting Moria

[Lord of Angmar]

Then again, he could always take the endless stair, walk to the top and climb down the mountain. The Fellowship tarried in Lorien for some time...

Well, Gollum caught up to them on their first night on the flanks of Lorien, and nearly found Frodo and the hobbits lying in a tree, before the orc band came. Besides, I doubt Gollum would be able to find or take said stairway.

My guess is he went ahead of the Fellowship, crossed the bridge, and awaited their departure from Moria to resume his tracking.

-Angmar