FORTUNE -- Dear Annie: I'll be starting my senior year in college in a few weeks, and right now I'm about two-thirds of the way through an internship at a company where I'd really like to work after I graduate. A counselor at my university's career center told me that it's important to show enthusiasm, so I've been coming in early, staying late, and volunteering for extra work after my regular assignments are done. Apart from that, is there anything else I can do to get hired?
Also, I recently got a small tattoo on the back of my neck, but then I started wondering, was that a mistake? My boss, who is a buttoned-down type of person, hasn't said anything about it, but do you think it could hurt my chances? --Kim
Dear Kim: Hiring of new grads overall is still sluggish despite the so-called recovery but, as you may already know, an internship gives you an inside track. Companies this year are extending full-time job offers to 63.3% of their interns, reports the latest survey by the National Association of Colleges and Employers, and end up hiring 53.3%. (Not everyone accepts the offer.)
"There are really four main things you need to do in order to turn an internship into a job offer," says Peter Handal, chairman of executive development giant Dale Carnegie Training. "First, be knowledgeable about the company and the industry. Second, be proactive. Don't sit there like a bump on a log, just doing what you're told. Instead, try to spot challenges and suggest solutions. Third, be creative. It will get you noticed. And fourth, be a team player. It's not just about you. It's about 'us.' "
Talkback: If you've turned an internship into a job offer, how did you do it? Leave your comments at the bottom of this story.
A common mistake, Handal says, is taking too passive an approach. "The transition from the classroom to the workplace is a real leap. In school, you're there to absorb information, period," he notes. "But in a business setting, absorbing information is only the first step. You then have to put your knowledge to work and show what you've learned."
He adds: "At school, you probably wouldn't ask a professor, 'Can I write an extra term paper?' or 'Can I take an extra test?' But at work, if you offer to do extra assignments and take on extra projects, your boss will be impressed. It will set you apart."
It sounds as if you're already doing that, thanks to smart advice from your college counselor, so here's another suggestion: Use your internship as the foundation for building a far-reaching network. "It's great to go out for a few beers or a ball game after work with your fellow interns and other people you work with directly. Those can be wonderful bonding experiences," says Handal. "But don't neglect to reach out and get to know other people at all levels of the company, including people several levels above you."
In addition, look for opportunities to meet as many people as you can outside the company. "Keep an eye out for trade association events, and even volunteer to help out by distributing flyers or whatever tasks need to be done," Handal says. "You never know who will notice you if you make yourself visible to people in the industry. Often interns are intimidated by the prospect of introducing themselves to executives -- they have the attitude, 'Oh, I'm nobody, I'm just an intern' -- but if you make yourself approach people and say hello, it can be a tremendous confidence builder."
Some employers, he notes, issue business cards to interns (bearing the title "intern"). Of course, networking isn't just about swapping business cards, but having a pocket full of cards does give you a graceful way of asking for someone else's, which could come in handy later. So if you don't have them, Handal says, ask for some. If it isn't customary at your company for interns to carry business cards, there's nothing wrong with a little gentle persuasion: "It certainly can't hurt to ask, since among other things, it's a sign that you are proud to be working there."
Now, about that tattoo: Without knowing more about the culture of the organization where you work, it's hard to guess how tattoos, piercings, and the like are perceived there. But it's safe to say that your "buttoned-down" boss may be more accepting than you think.
"Even in this tight job market, most companies are not going to view tattoos too harshly," opines John Challenger, CEO of outplacement giant Challenger Gray & Christmas. "One reason is that with everyone from soccer moms to MIT computer science graduates sporting tattoos, preconceptions about tattooed individuals are no longer valid. More importantly, companies have a vested interest in hiring the most qualified candidate."
He adds that employers might have a hard time finding candidates in your age group who don't have "some type of body embellishment": The Food and Drug Administration estimates that about 45 million Americans have at least one tattoo; and a recent Pew Research Center report on Generation Y (ages 18 to 29) found that 38% have tattoos, and Gen Xers (ages 30 to 45) aren't far behind at 32%.
If you're worried about higher-ups' reaction to your tattoo, you could always cover it up with a dab of makeup, or your hair -- or, once the weather cools off a bit, invest in some turtleneck tops.
Talkback: If you've turned an internship into a job offer, how did you do it? If you're a hiring manager, would a tattoo make you think twice about hiring someone, or is body art irrelevant? Tell us on Facebook, below.
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