Extending enum in derived classes

I have a class hierarchy, and each class in it has an exception class, derived in a parallel hierarchy, thus...

class Base
{
};

class Derived : public Base
{
};

class BaseException : public std::exception
{
   enum {THIS_REASON, THAT_REASON};
};

class DerivedException : public BaseException
{
    // er...what?
};

I would like, in the DerivedException class, to extend the enumeration type to include a new value THE_OTHER_REASON, so that the DerivedException class could hold any of the three values.

First of all, ought I to want to do this? Does it seem a reasonable practice? If so, how do I go about it? If not, what alternatives would you recommend?

EDIT: A possible duplicate has been suggested here, but the suggested solutions are different because that question is for C# and this for C++.