Reaching out to Headhunters?

Is reaching out to headhunters you have not heard from a realistic strategy? If so, should I just be sending an email to one of their associates basically saying that I am an analyst at ____ bank and would like to put myself on their radar / register with them / even jump on the phone? For context, I am an analyst just starting my second year at an MM and am starting to pursue off-cycle PE recruiting. I have heard from basically all of the big name headhunters aside from Dynamics and Oxbridge and would like to put myself on their radar if possible as I know they tend to have more coverage in the region I am interested in.