Should I Ask for a Job Interview?

I'm doing an out of town job search and the recruiter from one of my target companies finally called me to schedule an interview. I am paying for my own plane ticket to get to the interview. This is a large hospital so they have applicant tracking software that allows you to see what you status is.. (not qualified, under review, referred to hiring manager, etc).

I have several positions that have been "referred to hiring manager" but no interviews have been scheduled. Would it be bad form to ask the recruiter to inquire with said managers as to whether or not they might be interested in interviewing me while I am in town?

I don't want to screw this up, but it would be nice if I could meet with more than one department while I am there.

Should I Ask for a Job Interview?