As far as I know there is no greeting that corresponds to the English 'good afternoon'. I also think 'bonjour' is still used in this context. I think Bon(ne) après-midi is only used differently (that is, when leaving someone; e.g. je te souhaite un(e) bon(ne) après-midi). Is there a reason why this happens so? Why one cannot say bon(ne) après-midi even as a greeting (i.e when meeting somebody)?
I don't know if there is a good reason but:
are used when meeting someone while
- "Bonne journée"
- "Bon/Bonne après-midi"
- "Bonne soirée"
- "Bonne nuit"
are used when leaving someone
(I don't think I ever heard "Bonne matinée")
So you're right, there is no equivalent to 'Good afternoon' as a greeting and 'Bonjour' is usually used even in the afternoon. Note that I can only speak for metropolitan France.
There is no reason why you can't from the point of view of what the practice in itself represents — in itself this greeting is just as justifiable as "bonsoir" and "bonne nuit" —, but there are certain reasons why you might not want to do it, why you might not want to be a pioneer.
You can't say it and give people the impression that what you are saying is something that they can relate to, simply because nobody does it and so nobody is used to it. One of the problems is that given a usual greeting, people know how to react because there are phrases which are more or less set phrases that correspond to it and because they have been trained in various ways to do that. In the case of "Bon après-midi" there is the obvious reply that consists in saying "Bon après-midi" but as people do not recognize the greeting as one of those they've learned they are bound to hesitate or simply conclude that they don't understand well what you say or that simply you don't know too well and are making an error that you'll soon correct (in other words that you shouldn't say it) — although I think that on account of the rather obvious nature of the words some might reply in the way I suggest.
It is quite possible that after having used it with the same persons for a while, after a few occurrences of getting no reactions, some people should find it quite acceptable to answer you with a "bon après-midi". Here, again, the usage has to start somewhere, so why shouldn't you be the first? Nevertheless, there is the possibility that you should fail; you face then the situation of having to change your habit for another one (an accepted one), and this is not the kind of behaviour anyone can adopt lightly; rare must be those people with a certain philosophy of life who are undaunted by such a perspective.
You might say that you can't say it because people do not relate to it and by doing so you run the risk to create relationships that are bizarre and that in the end, if people are not responsive, you might be doing yourself much of a disfavour by displeasing them.