Jane, I don't think you hurt your chances by asking if they might have a position available. If you end up going out to one institute for an interview, call other nearby colleges and see if you could drop by and sit in on a seminar and talk with the chair if at all possible. Definitely talk with the faculty there; some of them might even have suggestions as to other facilities who may be in need of someone, soon if not at that particular time.

When you speak with the department chair's secretary/office manager, make sure to find out if there are any colloquia scheduled around the time that you would be visiting that part of the country. See if the department chair might be willing to meet with you, even if they don't have a job available there at the moment, so that you may introduce yourself as someone who is going to be part of the mathematics community in that geographic region since you'll probably land a job somewhere nearby. It never hurts to ask for advice; and if you want a job, it never hurts to let people know that you're looking. You may also learn a bit from talking with recent hires about the local mathematics community, and where spots might be opeining up nearby.