What’s the best way to call and ask for contact information for a hiring manager at a company?

Just ask the person who answers the phone if you can please speak with the hiring manager. If they put you through to a voice mail, you can hang up and call back a little bit later. Ask the receptionist when a good time to call would be, or if he/she has an e-mail address you can contact them through. You might end up having to just leave a voice mail though, or speak to someone in human resources, and not necessarily someone as high up as the hiring manager.
Good Luck.
If all else fails, write a letter. (Yes, an actual letter, not an e-mail).