Tailor both the resume and cover letter. How many times have you heard people say they customize their cover letter, but send the same resume to everyone? Usually they are very proud of themselves for doing this, as many job seekers send one form letter and resume to everyone. Unfortunately, when a resume is competing with 200 to 400 others, it has to stand out from the crowd. Recruiters don't have time to separate the "diamonds from the dirt" in the 10 to 60 seconds they spend skimming for relevant experience. It is the candidate's responsibility to sort the valuable stuff from the extraneous, using only 20-carat material to land an interview.

It's just a job. Grass grows, birds fly, waves pound the sand. I just beat people up
Muhammad Ali

Executive search firms need a good resume from you, but they also must have a search assignment that matches your background before they can be of service. Headhunters do not market you. They find the best candidates for client job openings. Their clients are companies that pay their fee or retainer. Don't expect a headhunter to make you her number one marketing project. Conducting your search campaign isn't her job, it's yours.