In that case it would be better if she was looking for Rukia. Maybe she's a forward scout for Byakuya and Renji. Its not like the Squads don't work together on certain things. Plus it means she has a reason to have to go back to the Soul Society when Rukia does. If Hana was in Karakura to keep an eye on Hollows she would have no reason to return until her assigned time period is over.