I'll be honest, publishing in the big journals isn't enough. It really is who you know and getting out there. Which is not necessarily natural terrain for a good number of academics. I understand this completely - I am hopeless at it and much prefer 'backroom' sort of work, if that makes sense. It's also arguably easier if one works at one of the more lauded institutions.
A good way in is to for a policy fellowship or get involved with anything that comes out of the cabinet office. Again, not for everyone. I have been know to suggest academics add in a colleague who loves this stuff to grant applications. They might be meh as a researcher, but if they can schmooze its a useful addition
Well, maybe I don't put it quite like that.