Broad advice I've seen for already-employed people looking for a change: Switch employers and roles at different times.

e.g., Learn enough to be useful, then talk to the security guys at your company. Prove you're useful and trustworthy; see if there's any tasks you can do for them without violating policy. If a spot opens up, see about changing roles within your company.

Or, join a smaller company, where your role and some security responsibilities overlap.

This is great advice and I've seen it work for a few people. (Not specifically for cybersecurity.)

The short version is when people can ease sideways to a completely new role at their current employer. The longer version involves getting a new job doing the same thing, but at a company that does a lot more of the thing you want to do. Initially you do the thing you were doing before, but there's more opportunity to shift sideways when you get a good rep.

This is some of the best advice!