Why can't the Voldemort's soul fragment in Harry find a new host

'Tell him that on the night Lord Voldemort tried to kill him, when Lily cast her own life between them as a shield, the Killing Curse rebounded upon Lord Voldemort, and a fragment of Voldemort's soul was blasted apart from the whole, and latched itself on to the only living soul left in that collapsing building. Part of Lord Voldemort lives inside Harry...'

Harry Potter and the Deathly Hallows - pp.550-1 - Bloomsbury - chapter 33, The Prince's Tale

What we have here, then, is a soul fragment existing outside of a body for long enough to "latch itself on to" another host, when its body is destroyed.

Why, then, can this not happen again? Why is the soul fragment destroyed in the Forbidden Forest, when Voldemort hits Harry with a Killing Curse? Why does the fragment of Voldemort's soul, blasted apart from its new host, not "[latch] itself on to [any] living soul" in the forest?

I answer this by saying that it is simply the nature of Horcruxes.

'But even if we wreck the thing it lives in,' said Ron, 'why can't the bit of soul in it just go and live in something else?'

'Because a Horcrux is the complete opposite of a human being.'

...

'Look, if I picked up a sword right now, Ron, and ran you through with it, I wouldn't damage your soul at all.'

...

'...But my point is that whatever happens to your body, your soul will survive, untouched,' said Hermione. 'But it's the other way round with a Horcrux. The fragment of soul inside it depends on its container, its enchanted body, for survival. It can't exist without it.'

Harry Potter and the Deathly Hallows - p.90 - Bloomsbury - chapter 6, The Ghoul in Pyjamas

But then I ask myself - well, if soul fragments can be blasted apart from the whole and latch onto other souls, why couldn't that happen again in the Forest? If Harry had been made into a physical Horcrux - like the others - I could sort of understand. A piece of soul is consciously transferred into an object, on which it then becomes dependent, because, of course, it's just a little fragment - not able to exist on its own, outside of its new container. But the soul fragment that latched itself onto Harry was not transferred by whatever the standard Horcrux creating spell is. More importantly, that fragment certainly was able to exist on its own and find a new body (much like the bit of soul that existed in the Forest of Albania during Voldemort's exile, and later had a body resurrected around it).

So, does anybody have anything more concrete to add on this. Hermione's explanation is okay, but it just seems a bit ... unsatisfying. I wouldn't say this is an out and out plothole, it just feels like the magic is ... arbitrary. And JK Rowling has generally taken great pride in making her magic consistent and making it make sense.

Answer

When the Killing Curse rebounded on the Dark Lord, it ripped him from his body and fractured a piece off of his main soul, and that piece went on to live in Harry as he’s a living host and souls that aren’t intentionally encased in a Horcrux object naturally only live in living beings.

“Tell him that on the night Lord Voldemort tried to kill him, when Lily cast her own life between them as a shield, the Killing Curse rebounded upon Lord Voldemort, and a fragment of Voldemort’s soul was blasted apart from the whole, and latched itself on to the only living soul left in that collapsing building.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)}}

At that point, the Dark Lord’s main soul was tied to life by the Horcruxes, so when it split, both pieces remained alive as the main soul cannot move on while there are Horcruxes tying it to life.

“Well, you split your soul, you see,’ said Slughorn, ‘and hide part of it in an object outside the body. Then, even if one’s body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged. But, of course, existence in such a form …”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)}}

The piece that fractured off was still part of the main soul until it did, so it was also tied to life then.

The second time the Dark Lord hits Harry with the Killing Curse, the soul fragment is already separate from his main soul, and therefore no longer tied to life by the Horcruxes. Since nothing tied it to life, it was simply killed when the Dark Lord’s Killing Curse hit Harry, and it in him.

“So the part of his soul that was in me …’ Dumbledore nodded still more enthusiastically, urging Harry onwards, a broad smile of encouragement on his face. ‘… has it gone?’

‘Oh, yes!’ said Dumbledore. ‘Yes, he destroyed it. Your soul is whole, and completely your own, Harry.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)}}

There was nothing keeping the soul fragment alive, so the Killing Curse killed it once it hit.