I've done this - with Web Development, but still got projects this way. I just responded to the recruiter or directly to the company if I had their info, and stated my background, my rate, and asked if they were interested in working with my in a contractor capacity. It's gone very well, and I have several ongoing contracts from this process.
Basically just state that you are interested in the work, but not the current agreement and see if the company is open to hiring you as a contractor. If your experience is good and viewable/provable, they probably will be. And Bonus points if you are a good communicator and timely with such and delivering deadlines - many contractors are not and companies/recruiters get burned by them and those who ARE dependable are worth their weight in gold to companies.
I always told them that i'm not looking full time but only part time but that only did work once I think. Most success came from building a small persona online, participating in the discussions, networking on Discords/Twitter and then the opportunities came.
Basically just state that you are interested in the work, but not the current agreement and see if the company is open to hiring you as a contractor. If your experience is good and viewable/provable, they probably will be. And Bonus points if you are a good communicator and timely with such and delivering deadlines - many contractors are not and companies/recruiters get burned by them and those who ARE dependable are worth their weight in gold to companies.