Almost everyone celebrates the new year by making resolutions. One of yours may be to rejuvenate your stalled career by seeking additional help from the circle of friends, colleagues, and acquaintances who can provide you with job leads, business introductions, and professional advice.

But beware! You may be alienating important contacts – who could shun your calls and bad-mouth you to others – if you ask for too many favors too often without reciprocating.

It’s possible to recover from network overuse. However, the process “takes place over time and is not an overnight cure,” says Susan RoAne, a Greenbrae, Calif., author of books about networking.

The best first step: Apologize humbly. Tell an over contacted contact, “I blew a relationship and I need that relationship,” recommends Diane Darling, a fellow author and president of Effective Networking in Boston.

Express gratitude for the contact’s generous past support, perhaps with a handwritten note tucked inside a belated New Year’s card. Make sure your next request better matches that individual’s capabilities.

“Don’t ask people for something so out of their realm that they feel uncomfortable and unable to respond,” Ms. RoAne advises.

One jobless marketing executive keeps pestering Ms. Darling for telecom-sector introductions even though the networking specialist previously supplied her a handful of industry acquaintances. “Why turn to me again?” Ms. Darling asks.

The executive has also ignored the suggestion that she tap her Wharton network. Fed up, Ms. Darling avoids her entirely. Unless you heed contacts’ counsel, they may refuse further referrals out of fear that you won’t represent them well.

To soften your pesky, self-centered image, transform a networking bond into a two-way street. Offer favors, information, or introductions tailored to an overused contact’s needs.

Joey Robertson lost his job last June as a Sara Lee purchasing agent in Winston-Salem, N.C. He stayed in touch with a human-resources manager there by phoning her every week. Through this friend’s efforts, he landed a few interviews – though no job – with a different department.

Three months ago, however, “she stopped taking my calls,” Mr. Robertson remembers. He left several voicemails. One of her associates finally told him, “She’s just busy. She’s not going to call you back.” He suspected his excessive appeals for assistance wore out his welcome.

But the 38-year-old resident of Pfafftown, N.C., hesitated to volunteer ways that he might assist the HR official. “I don’t have a whole lot to offer,” Mr. Robertson said in early December. He failed to realize that even unemployed job seekers have valuable expertise and connections.

Nonetheless, last week Mr. Robertson joined RF Micro Devices, a high-tech company in Greensboro, N.C.A coworker at a prior employer recommended him.

He had sent her his resume, then called every five weeks to check in. “I wasn’t bugging her,” he explains. “It took five months for that seed to come to fruition.”