Since I knew I was taking this position, I've received three more calls from different recruiting agencies. I've lost count of how many agencies already had permission to refer me for interviews (it's less than 10. I think.). Now that I'm employed, I won't be needing their services. Immediately. But even if this wasn't a term-limited position, all good things must end, and it would be stupid to throw all those contacts and associated goodwill away. So I'm going through my e-mail, letting each one know that I'm currently unavailable but appreciate their interest and asking to reconnect when my current position expires. The response has been entirely positive. One recruiter even asked if my new company had new openings they might hire her to fill. I can't say this fills me with confidence, but the recruiter who got me my current job didn't realize that there's a difference between compiled languages and scripts*, so I'm not sure it matters.
I should note that I didn't come up with this idea myself. I simply explained the situation honestly to the first recruiter that called after I had the job, and he asked if I'd mind if he kept me on file. I'm not sure it would have occurred to me otherwise, but it should have.
*Note to anyone who's here for the teaching-job blogging: if you were a programmer, and I'd delivered that joke better, you'd be shooting soda out of your nose with laughter. And the soda would be free. But you wouldn't get summers off.