Yes, getting a new client takes a lot of time, and you don't always know exactly when one will end, so it is hard to schedule another one to start until you are definitely done with the old one. In may case, I have several clients and I don't spend 100% of my time with any one of them, but your situation is very different because you are "supplemental staffing" which means they need you the bulk of the time.
In my last experience with independent consulting, back when the economy collapsed due to 9/11, it was hard to find any job, contracting or otherwise. Even if you tried to go back into full time employment, recruiters and companies treated former consultants like red-headed stepchildren. How DARE you go out and try to earn what you are worth! As punishment, you can't have a job with us. The job postings would even say stuff like "corporates only", "must have corporate culture", that sort of thing. That's probably illegal now, but back then if you did consulting and then tried to come back into a full time employee role, you were blacklisted.