Clearance for consultants

I have TS/SCI and want to leave my job with a big defense company to join a consulting company. The consulting company wants to use me to consult with companies that are developing proposals for TS/SCI work. The consulting company says they can’t hold my clearance because I am not a full time employee and they don’t have a contract. They also say that the companies who could use me will require that my clearance be passed by visit request from the company that “holds” my clearance. Is that true? If so, then it sounds like I am out of luck because when I quit my job no company will be “holding” my clearance.
I think that my clearance is with the government, so I don’t understand why a company has to “hold” my clearance.