The beginning of The Dispossessed isn’t a carryover from any longer works, and at the time it was published it wasn’t a carryover from anything. I read all the ’60s and ’70s Hainish stories in the order they were written. The Dispossessed takes place before any of the other longer works. The short story The Day Before the Revolution is kind of a prologue to it and came out the same year, but it was published a few month later. Le Guin probably couldn’t have controlled when the story was published in relation to the novel, so I don’t which order she wrote them in.

From Wikipedia:

In the first three novels—Rocannon’s World (1966), Planet of Exile (1966), and City of Illusions (1967)—there is a League of All Worlds; by City of Illusions, the League seems to have been conquered or fragmented by an alien race, the Shing, from beyond the League. In the fourth, The Left Hand of Darkness (1969), the planets of the former League have reunited as the Ekumen, which was founded by the Hainish people. The fifth, The Word for World Is Forest (1972), part of the anthology Again, Dangerous Visions (and only published as a separate book in 1976) is set before any of the first four books, and in it the League of All Worlds and the ansible are new, and the term “Ekumen” is not used. The sixth, The Dispossessed (1974), is the earliest Hainish novel, chronologically; in it, the Cetians have been visited by people from other planets, including Terra (Earth) and Hain, and while the various planets are separate, there is some talk of a union (and the ansible concept is known, but none yet exist). The seventh and final novel, The Telling (2000), and the later short stories only speak of the Ekumen—which now includes the Gethenians, who were the subject of The Left Hand of Darkness—and not of the League

Le Guin often discounted the characterization of a “Hainish Cycle”, writing on her website that “The thing is, they aren’t a cycle or a saga. They do not form a coherent history. There are some clear connections among them, yes, but also some extremely murky ones.”[1][2]

Le Guin offers the following thoughts on the order in which readers should approach the series:[3]

Rocannon’s World, Planet of Exile, City of Illusions: where they fit in the “Hainish cycle” is anybody’s guess, but I’d read them first because they were written first. In them there is a “League of Worlds,” but the Ekumen does not yet exist. / Then you could read The Word for World is Forest, The Left Hand of Darkness, [and] The Dispossessed, in any order. In Dispossessed, the ansible gets invented; but they’re using it in Left Hand, which was written fifteen years earlier. Please do not try to explain this to me. I will not understand. / Then in the collection of stories A Fisherman of the Inland Sea, the three last stories are Ekumenical, and we even finally find out a little about Hain, where it all began. The story suite Four Ways to Forgiveness is part of that universe, and so is the novel The Telling. But I have to warn you that the planet Werel in Four Ways is not the planet Werel in Planet of Exile. In between novels, I forget planets. Sorry. / The Eye of the Heron may or may not be set in the Hainish universe; it really doesn’t matter. As for The Lathe of Heaven and Always Coming Home, my Terran science fiction novels, they definitely don’t exist in the same universe as the Hainish or Ekumenical books.[3]

I assume that the claim Left Hand was written “fifteen” years before The Dispossessed is a typo, and she meant “five.” It was published five years earlier. I don’t think that she wrote it, at least not in anything like its final form, in the ’50s before she’d published anything. There are numerous reasons to believe that’s implausible.