Jobs at small Birmingham companies nobody posts about - 6 weeks in

A friend who works at a local manufacturer says there are openings not advertised online. It has been a long 6 weeks. They like a brief email with a résumé. What do you write in a cold email?

Thanks for writing this out. I went through something close to it 9 months ago, and it got easier once I kept a simple log of every call and form.

Visiting in person worked for a trades job. For office jobs, email was better.