Moving up within an organization is not an unrealistic expectation. Particularly if you have put in the time to make a contribution to the business or to improve the way something is done.
What can you do to improve the chances or getting that promotion you may want? The first thing to do is to stop thinking about it as it shouldn't be your primary concern.
The next thing to do is determine how to maximize the time that you spend at your work. Figure out a way to create more value out of your 40 hour work period. This does not mean that you need to work harder. Rather, think creatively about the work you do and figure out how to be more effective. Making more happen with less effort might be a good way to approach it.
Another way to get the attention of leadership is to help the company grow internally, externally, or both. Can you develop a new program that helps the development of new or current employees? Is there a more effective way to conduct meetings, or perhaps eliminate them? Can communications across the organization be improved? Is there a new tool that could be introduced that would make someone's job easier?
External development of a company involves the creation of new markets, or new programs and services to offer clients. This might also be described as expansion. Is your work with an organization contributing to growing the revenue of the company in some direct way, or helping to reach into new areas?
Related to both of internal and external development is the idea that you carve out a space of your own within an organization. When you make yourself the expert on a given topic--and can demonstrate the value of your work--you will create demand for your time. You will also be unique.
Regardless of how you approach it, moving up within an organization really involves helping a company move out of it's current state of business and into another. For you, getting there requires moving out of your comfort zone and into uncharted waters as you help a company explore what's next. Whether you work to improve a company internally or help it expand is dependent upon your interests, time, and resources available to you. In addition, you'll likely be doing your current job in addition to figuring out how to build your next one.