Apart from being an academic, what can a Discrete Math PhD do?
Short answer: Everything.
Mathematicians are often not (solely) hired for the graphs and algorithms they know, but for the skill to logically dissect a problem, come up with a solution, consider all possible cases, etc.
With discrete math, a career in software development would be possible for example. Look for R&D departments or positions and they might prefer a PhD with strong background and some coding skills over someone who has developed for years but is not able to turn a Wiki article on some algorithm into proper code because the math is lacking to understand it.
If you don't mind extending into statistics a little bit, data science and machine learning would open up, where also a lot of mathematicians are wanted currently.
The problem solving skills I mentioned above are also often liked by consultant agencies, where they care not so much about what you know (as you will need to learn and use their products either way mostly) but rather that you can take a client's problem and solve it. Also these guys really like PhDs, as that looks cool and they can bill you out higher than someone without one (even if your PhD is not that relevant to the client).
So, medium long answer: There are a lot of options, you simply need to get out there and look for them. You will need to do some convincing of the form "yes, I am missing these two skills you require, but I can more than make up for it by ...", but then you should be able to get into many fields.