A lot of you won't like it, but it's all about learning how to play the game:
-Choose to work on high-visibility projects where possible. It can be as simple as finding out about a new initiative and saying "hey boss, I heard about XXX, can I help out or tag along to meetings to learn?"
-Make sure your work is visible. Take meeting notes or create a scorecard to track projects and circulate them widely. Make sure your boss knows what you are working on, and when you have hit a milestone or a roadblock. Keep him/her in the loop and away from surprises. Err on the side of giving more people credit rather than less "We worked on..." even when it was just "I worked on." If your boss is good he/she will admit it was mostly you, but will appreciate the credit anyways.
-Volunteer for BS stuff like planning committees or round tables or whatever. Put yourself in positions where senior people get to know your face and name.
-When you get invited to a meeting with high level people, ask a question. "Hey, I may be off topic, but I know XXX group is working on YYY, will that affect this project?" Something like that. Don't present yourself as an expert, just offer up something related. Worst case is you get the answer "no that doesn't really apply here" or something, but now you're the person in the meeting, paying attention, and trying to connect the dots.
-Network network network. Attend the networking events. Go to town hall/leadership meetings instead of calling in. Ask a senior person to go to lunch to talk about your career or his/her role or career advice or whatever. Try to take advantage of cross-functional teams to connect with other people in the org you don't work with every day.
-Have something on your resume that tells people you are interested in your career and moving it forward. Advanced degree is the usual option here, but could be professional certification, membership in professional group, etc etc. Doesn't really matter what it is, but it tells people "I'm doing more than just the bare minimum to further my career"
-MOST IMPORTANTLY, make sure people know you are interested in moving up/around. Bring up next steps in performance reviews. Ask a manager about a role open on his/her team, or apply for it. You never want to be in a spot where you are told "oh, we never thought of you for that role, I didn't think you were interested." Even if you aren't interested, think about poking around and tell those involved "I'm just trying to understand if this is a good next step for me." Just the fact that you're thinking about a next step and letting people know it puts you ahead of the game.
-Finally, know when to cut your losses and bolt for the next company. Know what your potential next opportunities are and the timeline for when they will come available and how you rank against those competing for them. If they aren't obvious, ask. If they aren't there, GTFO. The grass isn't always greener, but it usually comes with more money and nothing has to be permanent.