If I target a position coach from another team, how do I make a proposal to that coach for a promotion?

Ex: I identified a LB coach from another team and I want him as my HC

I have played around with the system, the only coaches that I'm able to hire are unemployed or to promote my own coaches.

You can poach another teams' coaches only during early free agency. Once the final stage of early free agency begins (i.e. the draft), the coaching staff is safe (with the exception of them being fired by hiring an unemployed coach).

Thanks