Here is what I think, working for a big-name company :)
While it is true, that big-name companies will certainly open up doors and give you a lot more knowledge on your way (more people thinking about stuff and implementing it leads to constantly learning new things about coding, project management, etc.), I think small companies will give you the experience instead. In a big company, you usually have a limited field of responsibilities, but you can specialize in one. In a small company, however, everyone counts and there are usually not enough people to do all the tasks, so everyone has to do a bit more. That leads to broadening your horizon! Something like that can be a door-opener, too, if you just use the argument the right way :)
Apart from that, a big company can offer you added value by connecting you with internal specialists from all over the world, send you to some foreign country at no cost for you at all. Additionally, when staying with a big company, they will give you bonus values, like a retirement arrangement, medical care (we have a private doctor on premise), a multitude of different jobs and positions if you ever want to try out something new, and so on. Small companies can most of the time only offer familiarity, but that can be a lot more important depending on your personality and that of your co-workers :)