I understand the situation, and the key is to be brief and specific. First, if you can get your former boss to make an introduction, that would be the best way in without question.

If that’s not possible, start your message to them identifying the commonality you have with them, that your network on LinkedIn lead you to them, and that you’re also in the print manufacturing industry, also that you run in the same circle of people. If you can name drop your former boss, if that is appropriate, then do it, but then quickly transition into showing the strong interest you have in their company, and you were hoping they might be willing to share a couple thoughts about the company and why they like it there; perhaps things that wouldn’t be obvious to people who don’t currently work there.

Don’t ask for an interview or where to apply or how to pursue a job with them, yet. The first contact should be about getting their opinion about the company and sharing your interest in it with them. You’ll be more likely to get a response. Then when they do respond, then ask about who you should talk about positions, or the best place to apply. Taking that approach will help you build rapport first, then you can get the info you need to leap over the typical online application to get moved to the front of the line. Does that make sense?

Once you have the message you want to send together, please feel welcome to post it here (omitting names of course), and I’ll review it and offer any suggestions I can that would be helpful. Thanks for the question!