I feel like Sato, being overly possessive and all, would be put off by anyone else having anything to do with Shio. So I imagine that she gave an old sock of her own or crafted one similar enough to trick him. I wouldn't even be surprised if she smelled just like Shio at this point. If Sato is crazy enough, and the boy isn't crazy enough to magically know if the scent belongs to Shio, Sato could use her scent on the sock to get him familiar/addicted to her scent/existence instead as he keeps indulging himself with the sock (if they don't smell 100% the same, that is).
But that's a convoluted plan so maybe she just sacrificed the sock after weighing the pluses and minuses of getting him knee deep into trouble with her. It's manga so if people are crazy enough, they'll immediately recognize the scent of the people they're obsessed with by manga law even if they haven't been in contact with them for long periods of time. I hope she didn't sacrifice the sock though!