I'm writing in second-person just because its easy, for you.
You are working with a game engine and really wish a particular engine class had a new method that does 'bla'. But you'd rather not spread your 'game' code into the 'engine' code.
So you could derive a new class from it with your one new method and put that code in your 'game' source directory, but maybe there's another option?
So this is probably completely illegal in the C++ language, but you thought at first, "perhaps I can add a new method to an existing class via my own header that includes the 'parent' header and some special syntax. This is possible when working with a namespace, for example..."
Assuming you can't declare methods of a class across multiple headers (and you are pretty darn sure you can't), what are the other options that support a clean divide between 'middleware/engine/library' and 'application', you wonder?
My only question to you is, "does your added functionality need to be a member function, or can it be a free function?" If what you want to do can be solved using the class's existing interface, then the only difference is the syntax, and you should use a free function (if you think that's "ugly", then... suck it up and move on, C++ wasn't designed for monkeypatching).
If you're trying to get at the internal guts of the class, it may be a sign that the original class is lacking in flexibility (it doesn't expose enough information for you to do what you want from the public interface). If that's the case, maybe the original class can be "completed", and you're back to putting a free function on top of it.
If absolutely none of that will work, and you just must have a member function (e.g. original class provided protected members you want to get at, and you don't have the freedom to modify the original interface)... only then resort to inheritance and member-function implementation.
For an in-depth discussion (and deconstruction of std::string'), check out this Guru of the Week "Monolith" class article.