Disclaimer: Jin is really cool and i'd love to see him involved in things outside of GhoT
One day Jin Sakai finds himself following a strange white furred animal while on the hunt for Mongols who'd been sighted in the area terrorizing the local wild life.
Tis a strange thing this animal, making noises more in line with a human attempting to mimic the call of an animal rather than an actual animals call, but even stranger is the fact that the moment he turns head to take in his surroundings and look for any landmarks he could identify the strange animal disappears and he is suddenly enveloped in a thick fog that makes his poison feel like food poisoning in comparison.
After what seems like an eternity, Jin suddenly finds himself face to face with a young blonde haired, green eyed child surrounded by what was obviously a band of thieves.
Every breath he takes makes him feel like he's breathing steel into his lungs, but even still, there is a child in need of saving, and The Ghost has no choice but to ignore his pain and protect them, even if it ends up killing him.
Jin Sakai winds up in the Fate universe right after Artoria pulls Caliburn from the stone but right before she apprentices under Merlin, so like the day after she pulled it out. I'm assuming Merlin gave her a day to pack the necessary provisions before having her meet him somewhere else. The "White Furred Animal" is indeed Fou. Jin's having a real tough time acclimating to the massive amounts of mana permeating the world, dick wizard might wanna do something about that if he wants Jin to live for more than a day.