The first thing to remember as you’re striving for a promotion is that you need to figure out how to get noticed for higher level skills -- phones and calendar management are no longer relevant once you've moved up. Look at the job responsibilities of the person one level above you and start taking some of them on (this could work out very nicely if that person is also trying to get promoted -- she may be willing to hand off some of her duties to make room for her own higher level assignments). Find things that are outside the realm of administrative duties and learn to do them well. Maybe you’ve never been asked for your thoughts on a script or project, but that doesn’t mean you can’t share your opinion. At the very least, you could read/watch the content and have your notes ready to go in case someone asks for them. But if you have a really strong point of view and have cultivated your taste by reading and watching a lot of content, you may be able to impress a higher-up by offering your feedback in a humble way, even if you haven't been asked. Just keep in mind that you shouldn't offer unsolicited feedback until you have established a very solid rapport with your supervisor.
Secondly, you should find ways to take initiative. If you see a problem with the way things are run on your team, fix it. This could range from implementing some type of new organizational process that improves efficiency to generating a competitive report that will allow your department to develop content that will stand out in the marketplace. You could even find new projects or talent to bring in -- no one is going to stop you from getting coffee with potential writers or directors that could help make your team’s product stronger. If you can come up with a list of concrete accomplishments that are the result of you taking the initiative to get something done on your own without being asked, it will be hard for your boss to argue against a promotion.
As you develop yourself professionally, remember: You CANNOT let your current responsibilities slip. Your work as an assistant is essential to the daily functioning of your department, so administrative errors will be more noticeable than any achievements you may be making outside of your pre-defined role. Wait until your assistant duties have become second nature before trying to take on extra work. Your boss will recognize that you’ve put in your time and that you’re capable of handling a heavier workload, and only then will you get your promotion. And if there's no room for growth at your company, you can always position yourself to level up at a different company with your newly cultivated skills.