One of the best ways to grow your fan page organically is to post VIRAL content frequently.
BUT in that case, you really need to know EXACTLY who your audience is and what triggers them.
Spend a little time on your competitors pages and see what is going viral on their pages.
You can also research viral content on Buzzsumo page (free version can get you started)
Post 3-6 times per day, and you will organically start growing your fan page.
Once sometimes goes viral, you might do a boost post to get even more exposure and boost post is really cheap.
SIDE NOTE: For example, if I was growing an audience for NON GMO type of crowd, I know that organic food, organic movement, would work well to get posts to spread. So you need to know that in order to grow it.