What is the best way to make a connection, develop an "in," and ultimately get work as a freelance book designer/page layout artist with the big publishers, like Bertelsman, Rowman & Littlefield, Harcourt, HarperCollins, etc.?

Let them know who you are. Send them your portfolio - your motivation why you want to work for them (...don't mention 'paycheck' though =]...), create muck-ups of things they already published in the past, etc...

Yes, I certainly realize I need to let my prospective audience know I'm out here, who I am, and what I do. Toward that end, twice each year for nearly 10 years now, I send out a cold email to every publish in the current year's Writer's Market that I can locate an email address for--better still if I can find a specific person (a Production Supervisor or Design Director, for instance) to direct this email to. The body is sort of an electronic cover letter and I often attach work samples.

I've alternated back and forth with the attachment, trying to see whether I get a better response with or without. Surprisingly--or maybe not so much as wider bandwidth and speeds have become the norm--I've gotten a better response with the work samples attached. In fact, in the nearly 10 years I've been emailing my work solicitation, I've had only one case where someone expressed annoyance that I'd sent an attachment.

Additionally, I now have a website that presents me and carries both work samples and testimonials from people for whom I've already designed and/or laid out books. I also have a blog to discuss freelancing, publishing, book design, etc.

What I'm looking for is something I haven't thought of--when it comes to snail mailing samples, package, phone calls, and in-person interviews ... well, been there and done that. I'm looking for something new that will get me to the right people at these large publishers.