[FS]

It's been some time since I read Return of the King, but I believe he does. Because:

* * * SPOILER TO THOSE NOT FAMILIAR WITH THE STORY OF THE TWO TOWERS BEYOND THE MOVIE * * *

Near the end of the Two Towers, Frodo gets stung by Shelob and Sam thinks he's dead. He decides to complete the quest of the ringbearer, puts Frodo's ring around his neck, and leaves. But he's waited too long, and orcs show up, so Sam slips on the ring to become invisible.

Even though he had it on only for a moment, everyone who's ever worn the ring must eventually go to the Grey Havens and leave Middle-Earth. However, if I remember correctly, Sam doesn't go to the Grey Havens for many years after people like Gandalf, Bilbo and Frodo have left.