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Counterpart to headhunters

As others have said, some headhunters will work with you to help you find a good position. Unfortunately, in my experience they are more concerned with filling positions than with whether the person filling the position will be happy with it.

There are companies that help a job seeker find a job that are paid by the job seeker. I have no personal experience with them and they seem less common than a headhunter paid for by the organization trying to fill a position. Unfortunately, they don't seem to have a good reputation in general. One problem I've heard of is that they take up a pretty large chunk of your first year salary. Another reason is that they usually don't have the connections to really help job seekers find good positions. However, there can be some good ones.

In the United States at least, many colleges and universities have offices that help find jobs. While their primary focus is usually on those who are about to graduate or have recently graduated, some will help with alumni for at least a while - I believe my college helped for up to 5 years after graduation.

Some communities also have people who help job seekers find work. Check your local government website for such services.

For all of this, you will likely need to develop a good relationship with whoever you are working with. They are more likely to help you if you can make it clear what you want. When you have a more personal relationship with them, they are more likely to want to work to help you.