How do I approach contacts when I'm job searching?

Directly and honestly — but not like a supplicant. Bad: 'I'm looking for a job, do you know anything?' Good: 'I'm currently open to new opportunities in the field of X. Do you have 15 minutes for a brief exchange about the market? And if you know someone who might be relevant, an intro would be very valuable.' Start by giving — offer value first before asking for something.