I agree with networking. Also, making sure other people can do your current job. If you're the only person who can do what you currently do, then it won't be possible to move you on, because they'll need you to do your current job. So make sure things are documented, and that you offer training to others.
It depends where you are, too - I heard from someone from a company last week, who said they have very clear criteria for each grade, and if you meet the criteria, you get promoted, end of story. That is similar to my place in that we have very clear criteria, but it's also dependent on budgets and requirement in the team (you can only have so many senior engineers in a department), and recommendation from your manager (which is where I'm falling down, and likely to be raising a grievance, so they at least review my suitability for promotion, rather than an outright no without even looking at the evidence I've put together.)
Try to get on any big projects going round - make sure your manager knows you want more of a challenge and emphasise your readiness - your experience and skills (you think they will remember - they don't always.) If other opportunities come up, make the most of them, such as organising some documentation, or reviewing a process to make it more efficient or save money. (Anything you've done which has lead to efficiencies in time or money or made more money should always be emphasised, showing you benefit the bottom line.) Speak up in meetings, and be the first to speak up.
Are there any opportunities for voluntary things at your place? I've recently organised an event, which did take lot of my time outside of normal hours, but it's had really good feedback, and made me very visible to the wider organisation. There is also an internal magazine which I've written an article for, and at another internal even recently, one panellist dropped out through sickness, and I stepped in (it was a subject where I knew my stuff, mind you.) That session was being videoed, and will be viewed by a lot of other people round the company. So you have to make the most of every opportunity which comes your way (there are limits, on account of there only being 24 hours in a day, but sometimes, you just have to make the time, if it is a really good opportunity.)
We have talks from various leaders high-up in the company, and one of the things which is particularly notable is that few of them have gone straight up a normal ladder career path - most have gone sideways and even backwards at some point, to gain extra experience in an area they don't really know, or because the opportunities there were much better than where they currently were.
And get your CV up-to-date and start looking around for what else is out there, because it might happen they're just not going to support you, despite your best efforts, and in that case, you might as well move on.