Instead of asking someone if he has heard of any job openings, say: "Let me know if you hear of anything." The difference is subtle but important because it takes the pressure off the contact.
Do not ask anyone to distribute your resume - your job search is your responsibility. If a contact is interested, he will offer to do so without any prodding from you.
When you call a new contact and get voice mail messages, do not ask the person to return your call. Instead, close with: "Sorry that I missed you, I'll try again. If you would like to reach me earlier, feel free to call me at..." This gives the person an option of whether to call or not.
Do not limit your network of contacts - avoid being exclusive about who you admit into your sphere. Expand your network to include professionals at varying experience levels, not just your peers. Look outside your industry: you never know who might have the right connections. Ensure you make status updates and use professional-looking photos - not holiday snaps - online.