There's nothing wrong with reaching out to the CISO and be direct: "I work in xxx but would really like to transfer into your area. Can I grab a few minutes with you some time. Coffee on me."
The other thing for building your network is find your local ISSA and (ISC)2 chapters plus any other groups. Try to get to some meetings, and network that way.
If you reach out to people it may be better to ask for their advice on how to enter the field rather than ask them straight out for a job. You could also ask if they know from their network any other organisations that may be looking to hire less experienced resources or someone with transferable skills.
In the past I've approached internal audit departments, as they will know if there's slow progress on addressing audit findings and potentially if these are due to resource constraints. If you're lucky you might get a lead on an opening before it's even advertised, but you have to put in the leg work.
Absolutely, honestly networking, possibly getting some chance for mentorship or even job observation if allowed and maybe putting a face to the name if an appropriate role opens up in the future would be my main goal for this. That is a great idea though about asking about other organizations; I'd even be willing to come in on weekends and help with intern level tasks or just anything menial if it meant experience.