Disclaimer: my answer is based on the literary works by G.R.R. Martin, not on the TV series, that departs from the books lore by not even explicitly defining the true identity of this character; I watched and enjoyed the show, but it oversimplifies many things and render some of them rather confusing and inconsequential, that's why I'm sticking to the books here
I think that the basic premise of your question is based on assumptions that don't have many facts that support them. Even if on a first impression one could be lead to think so, there is no evidence that both "Three-Eyed Crow" and "Greenseer" are titles or positions to be held, that are passed form one person to another, in the same fashion of nobiliar titles, religious hierarchy or other titles granted to leaders (like, in example, the King of the North, the High Septon, the Lord Commander of the Night's Watch, and so on).
We can gather informations about Greenseers mainly from Bran's chapters of the novels:
"They were a people dark and beautiful, small of stature, no taller than children even when grown to manhood. They lived in the depths of the wood, in caves and crannogs and secret tree towns. Slight as they were, the children were quick and graceful. Male and female hunted together, with weirwood bows and flying snares. Their gods were the gods of the forest, stream, and stone, the old gods whose names are secret. Their wise men were called greenseers, and carved strange faces in the weirwoods to keep watch on the woods. How long the children reigned here or where they came from, no man can know.
A Game of Thrones - Bran VII
"Then you teach me." Bran still feared the three-eyed crow who haunted his dreams sometimes, pecking endlessly at the skin between his eyes and telling him to fly. "You're a greenseer."
"No," said Jojen, "only a boy who dreams. The greenseers were more than that. They were wargs as well, as you are, and the greatest of them could wear the skins of any beast that flies or swims or crawls, and could look through the eyes of the weirwoods as well, and see the truth that lies beneath the world.
A Storm of Swords - Bran I
"Only one man in a thousand is born a skinchanger," Lord Brynden said one day, after Bran had learned to fly, "and only one skinchanger in a thousand can be a greenseer."
"I thought the greenseers were the wizards of the children," Bran said. "The singers, I mean."
"In a sense. Those you call the children of the forest have eyes as golden as the sun, but once in a great while one is born amongst them with eyes as red as blood, or green as the moss on a tree in the heart of the forest. By these signs do the gods mark those they have chosen to receive the gift. The chosen ones are not robust, and their quick years upon the earth are few, for every song must have its balance. But once inside the wood they linger long indeed. A thousand eyes, a hundred skins, wisdom deep as the roots of ancient trees. Greenseers."
A Dance with Dragons - Bran III
From the text, we know that Greenseers were exceptional individuals, in the sense that they were born with abilities that nobody else had, and because of this they seem to have been chosen as the leaders of their people. So, in a way, Greenseer was just an attribute given to a particularly blessed individual, pretty much like in the real world we could call "genius" someone like Albert Einstein.
It also had the connotation of a position of leadership, but it seems that they acquired this thanks to their abilities, and there is no proof that only one Greenseer lived at a certain time (the texts seem to imply the other way around), or that there was something like a supreme leader like a organized religion could have. They seem to be more like druids and tribal shamans rather than a formal hierarchy of priests.
Going on with the example above, we can see that real geniuses are not common even in our world, but we don't have a genius "position" that is filled when the previous one dies; and of course both Einstein and Turing were without a doubt geniuses, and they lived during the same timeframe; I think that we can safely speculate that with Greenseers it was something like that.
About the Three-eyed Crow. And Brynden Rivers too.
We know that Bran sees a Three-eyed Crow in his dreams; there is not much reason to quote every occurrence, it is well established without the need to provide references. It is more interesting to note that even Jojen Reed saw it in one of his green dreams:
Jojen's eyes were the color of moss, and sometimes when he looked at you he seemed to be seeing something else. Like now. "I dreamed of a winged wolf bound to earth with grey stone chains," he said. "It was a green dream, so I knew it was true. A crow was trying to peck through the chains, but the stone was too hard and his beak could only chip at them."
"Did the crow have three eyes?"
A Clash of Kings - Bran IV
We can see here that these green dreams are made by metaphorical images that have to be correctly interpreted. Bran never thinks about himself as a Winged Wolf, it is just how Jojen sees him in his dreams; a similar thing could be said for the Three-eyed Crow, it is supposed to be just the dream image of someone else, even if we, like Bran, initially don't know who he could possibly be. (also check this question).
Other characters that don't experience green dreams, very likely speak of it just because he listened to him telling his dreams, there is not a single piece of evidence that they have previous knowledge about a Three-Eyed Crow.
When Bran & company arrive at the cave, they met a cadaverous man trapped inside a tree; we know that even if it is not explicitly stated, he is Brynden "Bloodraven" Rivers, I won't go into detail here; just like Jojen was able to tell thet his Winged Wolf was Bran when he saw him, Bran was able to recognize his Three-eyed Crow; it is interesting to note that Bloodraven seems surprised to be called like that:
"Are you the three-eyed crow?" Bran heard himself say. A three-eyed crow should have three eyes. He has only one, and that one red. Bran could feel the eye staring at him, shining like a pool of blood in the torchlight. Where his other eye should have been, a thin white root grew from an empty socket, down his cheek, and into his neck.
"A … crow?" The pale lord's voice was dry. His lips moved slowly, as if they had forgotten how to form words. "Once, aye. Black of garb and black of blood." The clothes he wore were rotten and faded, spotted with moss and eaten through with worms, but once they had been black. "I have been many things, Bran. Now I am as you see me, and now you will understand why I could not come to you … except in dreams. I have watched you for a long time, watched you with a thousand eyes and one. I saw your birth, and that of your lord father before you. I saw your first step, heard your first word, was part of your first dream. I was watching when you fell. And now you are come to me at last, Brandon Stark, though the hour is late."
A Dance with Dragons - Bran II
We know that in ASOIAF, dreams and prophecies are never explicit, they are always in the form of images that have to be interpreted; Bran was seen by Jojen as a Winged Wolf because of his House's heraldic emblem, it was a symbolic representation of his identity; pretty much in the same way, Bloodraven was symbolized in the green dreams by a crow, bot because of his nickname (derived by the shape and color of his birthmark) and by his past as a member of the Night's Watch (they are called "crows").
As I have pointed out, he seems to be surprised to be called like that (he was seen by others in green dreams, but presumably he wasn't present in his own dreams in a symbolic form), and when he answers, he seem to acknowledge more his past as a member of the Night Watch ("Black of Garb") rather than the form he took in other's dreams.
As things stands right now, we don't know much about Bloodraven acts and motivations in the Night Watch:
Ser Brynden Rivers set sail for the Wall late in the year of 233 AC. (No one intercepted his ship). Two hundred men went with him, many of them archers from Bloodraven's personal guard, the Raven's Teeth. The king's brother, Maester Aemon, was also amongst them.
Bloodraven would rise to become Lord Commander of the Night's Watch in 239 AC, serving until his disappearance during a ranging beyond the Wall in 252 AC.
The World of Ice and Fire - The Targaryen Kings: Aegon V
We know that a very small fraction of the Children of the Forest possessed exceptional powers, and that those individuals were called Greenseers; this name was more of an "attribute", a "quality", rather than a formal title or a position to be held. This is further supported by the fact that we also know two examples of humans born with this ability, Bloodraven and Bran Stark: surely they weren't leaders of the Children of the Forest. They were Greenseers too, because of their abilities, not because they were elected, invested, or because somebody else left that "position" vacant; and You basically are born as a Greenseer, you don't become it.
In a similar way, the Three-eyed Crow was just a symbolic image of a specific person in some other people's green dreams, when directly called like that, Boodraven did not immediately seem to understand; even here, this is the only evidence that we can collect form the text, it is not stated anywhere that this could be a title, a position to be held, or anything like that. Certainly, nowhere in the books Bran was called Three-eyed Crow, not by Bloodraven handing down the "position" to him, nor by anybody else; he was just being acknowledged as possessing the powers of "a" Greenseer (not "the" Greenseer).
We don't even know if there was someone that "called" Bloodraven at the cave in the same fashion as he "called" Bran, or if he went there by his own free will (he was reputed to be a sorceror and a man of great knowledge after all), or even if he arrived there by chance.
Similarly we don't even know if what he is doing here is because he is fulfilling someone else's duty, if he is acting on is own initiative, or in the end why he is here at all, besides teaching Bran to realize himself (we see anything only from the point of view of the little Stark after all).
So, to answer your question:
Literally speaking, we only know that the Three-eyed Crow was a dream image symbolizing Bloodraven, tied to his nickname and birthmark and to his past as a Brother of the Night's Watch. It is highly unlikely that somebody else could be depicted in this way, even Bran was seen as a Winged Wolf. In a way, it was just a nickname like "The Hound" for Sandor Clegane or "The Blackfish" for Brynden Tully.
If we want to use the Three-eyed Crow in a figurative way to describe what Bloodraven is doing, there is no evidence whatsoever that there was really somone before him doing what he is doing now; the assumption that this could be a position that is handed down from an individual to another is just that, an assumption that is not confirmed (or denied, for that matter) anywhere in the books.