The best jobs in his field are in big cities like New York, and industry recruiters weren’t coming to his university. So, through the career website LinkedIn he started reaching out to people who worked at companies that he had an interest in. His first few introductions didn’t get much response. Mr. Perry acknowledges that his notes were cordial but may have sounded too self-serving. So he softened his approach.

“I’d put in the title that I was a recent M.B.A. graduate looking for advice,” says Mr. Perry. “I’d focus on talking about them during informational interviews and I’d mention any accomplishments of theirs that I may have read about.”

He also participated in LinkedIn industry groups that allowed him to communicate directly with contacts he couldn’t have reached without a chain of introductions.

One fellow group member worked for Reckitt Benckiser, a British consumer-goods company with a major office in Parsippany, N.J. The contact agreed to speak with Mr. Perry on condition that he not ask for a reference. After a pleasant conversation, the contact ended up forwarding Mr. Perry’s résumé to the HR department, which helped him get an interview and a job. He’s now a senior leader on Reckitt’s e-commerce team.

Social-media websites like LinkedIn have made job hunting easier by automating many tasks. But one-click networking invitations fall short when trying to reach people, say experts. Instead, job hunters need to engage other professionals on a more personal level. This includes getting introductions to people outside of your network who can help you with your career. The trick is knowing how to ask.

Focus on your first- and second-degree connections. The former are contacts that have already accepted your invitation to join your network, or vice versa, and the latter are contacts known to your first-degree connection. Third-degree connections require more than one introduction and can be difficult to reach, as you may not have a mutual acquaintance.

Consider whom you want to get an introduction from. Do research on LinkedIn, Google and even Glassdoor to find out what kind of employee your chosen first-degree contact is, says Dan Schawbel, a workplace expert from Boston and author of “Promote Yourself: The New Rules of Career Success.”

“You really don’t want an endorsement from somebody that’s not liked at work or who has a poor workplace reputation,” says Mr. Schawbel. “How much do you really know about that person that you connected with at that trade show?”

When asking for an introduction, make it easy for your first-degree contact by mentioning how you’d like to be introduced and the reasons you need help, writing out the introduction to be forwarded and providing something in return for the effort. Networking works best when both parties can offer the other something useful.

You should also give your contact an easy way out in case he or she isn’t comfortable making the introduction, says New York career coach Melissa Llarena. “Always be gracious, since at the very least they might offer useful advice or a referral to somebody else that can help you with an introduction,” she says.

Groups are one way to contact second- and third-degree LinkedIn members directly. But don’t join a group and start contacting individual members without making an earnest attempt to participate in community dialogue. You could get ejected from the group by the moderators.

Answer some questions and start new topics. Go beyond “liking” updates by making thoughtful comments on new posts or by sharing relevant links. You want to show regular engagement, says Mr. Schawbel. That includes answering any network requests promptly.

Preface any requests with regular status updates to show that you’re active. It will also give you and your contact something to reference. Just don’t share the same personal updates about your cats across all of your social-media accounts. Try to keep your LinkedIn updates on a more professional level.

You’ll get a much better response if your profile is up-to-date and includes a photo. That includes making your profile relevant to the job that you want to get, says Ms. Llarena.

Once you’ve established a regular dialogue, take the relationship offline as soon as you can. Ask for 30 minutes and treat your contact to coffee or lunch.