Apparently, it is my role in this project to make lists. Which is amusing, because I never do that in real life. But anyway....
Season 1: Do we have a TB cameo? There is potential for one in Episode 2, where the Ainur are gathered at Almaren for the first time. He could ignore the invitation and wander off. A more significant time for him to wander off would be during the debate after the destruction of the Lamps, but he's almost definitely not a character who was living at Almaren. Also, we have scenes of both Sauron and Oromë discovering the elves, and thus wandering through that part of Middle Earth. Either of them could see him or be seen by him from afar. It could be one of those 'is someone there?' moments and he vanishes. I have to go back and see if we decided on one.
Season 2: Episode 4, the migration of the Teleri. Olwë's party crosses (the future) Eriador, and sees someone in a blue coat.
Season 3: ??? Denethor son of Lenwë sees TB pulled into a river and he vanishes.
Season 4: ??? Men cross the mountains into Beleriand in this season. When Finrod reads their minds/memories, we could see a glimpse of our yellow-booted friend in the memories of Bëor's people.