I think some flavor text saying she's headed there would be good, if she's not staying there. However Keros warns about how dangerous travel is (hence the Torii gate at their home), and I can't imagine the Champion letting her walk their alone in the forest (though she's a damn cannon in battle so… who knows). Although I imagine she'd probably expect the Champion to join her as well in their home, so there's that too.
I know in one of Tobs's Kinu stories, Inari Kinu complains how her mother can go into town undisguised but she can't. Kiyoko replied that it was because the Champion was respected and trusted, and everyone knows she's with him. I kinda assumed that was what was going on. But I've been wondering about how the Champion is being perceived after transforming into a Kitsune. Maybe it's a power like the Foxfire weapon spell Nakano can teach, that eventually we'll learn to polymorph or disguise ourselves like other Kitsune.