If I remember correctly . . . (I've checked some of these, but not all)
Bilbo had The Ring on a chain to keep it safe, and put both Ring and chain in the envelope for Frodo.
Frodo unfastened the Ring from the chain before handing it to Gandalf in The Shadow of the Past when Gandalf tosses it into Frodo's fire.
When he's tempted to put it on when the Black Rider approaches, he never gets beyond pulling on the chain.
When Frodo gives the Ring to Bombadil, he pulls out the chain and unfastens it to hand the Ring to Tom.
Frodo retains both clothes and Ring in the Barrow -- everything the other Hobbits had on them is lost there.
When it slips onto Frodo's finger in The Prancing Pony, it may still be on the chain, which is attached to his belt, but nothing of this is mentioned.
In Rivendell, we're told the Ring gets a new chain, but not why. Presumably this is the "unbreakable" chain you mention.
I can't find "Sam's inability to break the chain" that you write of -- we're told he undoes the clasp and lifts Frodo's (apparently dead) head to draw the chain over it, which is confusing me a bit: one or the other, yes, but why both? This must be what's confusing you, too.
I gather from your spelling of "Kirith Ungol" that you might be reading a translation. . . here is the text in my copy:
The Two Towers said:
He stooped. Very gently he undid the clasp at the neck and slipped his hand inside Frodo's tunic; then with his other hand raising the head, he kissed the cold forehead, and softly drew the chain over it.
Aha! It was the clasp of Frodo's
tunic that he unclasped! It seems you are correct: the new chain may be permanently chained through the Ring.
In the tower of Cirith Ungol, Sam "passed the chain over his head" to give it back to Frodo.
But at the Cracks of Doom, no mention is made of the chain. Gollum bites off the finger and there is clearly nothing attaching The Ring to Frodo, though it might still have an unmentioned chain hanging from it. If Frodo still had the chain around his neck at the Cracks of Doom, he would probably have slipped it over his head to put the Ring on his finger. But we're not told.
I may have missed it somewhere, but I did not find an answer to your question!