A lot of responses to this question are more upbeat than I think is warranted. Many answers give detailed lists of uses of mathematics in a way that suggests the writer has no experience actually hiring people out of academia to meet those needs. There is an awful lot of "fields X, Y, and Z need people to do math, so you can probably get a job doing that." The irony is that this attitude is most common within academia. Most people who will assume that a math PhD with no job experience outside of academia is good for something, to the point of paying them to work on an applied problem, are in academia--- in bioengineering, machine learning, and other fields that people have recommended. The point I would underline here is that these people work in universities and if you want to get into these areas, you will have to stay in academia, at least a little.
The experience of a researcher at a university in some applied area is very different from the experience of a professor of pure mathematics. For example there are often fewer teaching duties (e.g. lab supervision, instead of teaching large classes--- or no teaching duties at all). And there are more options for sources of outside funding--- unlike in most of math, where if you don't get a grant from a government agency that funds math, you aren't getting a grant. But there will still be publishing papers, and you will still spend the majority of your time with people who share their social characteristics more with other academics than they do with the general population.
Academia isn't the only culture with negative aspects. If you look for a private sector job, you will find that most people--- even in very technical companies--- are not open to hiring people with no private sector experience and no personal connections for non-entry-level jobs. And they aren't open to hiring PhDs for entry level jobs (you are "overqualified"). At many companies, it's not "we do a lot of mathy stuff, so math PhDs can help a lot," but "we do a lot of real world stuff here, and anybody who spends decades buried in textbooks won't know anything about that." For example, unless you have an easily documented and publicized history of programming (e.g. contributions to open source projects, or reasonably self-contained projects that you can make public and stick on a personal website), most companies will not give you a second look for any software engineering job. Even if you've done a ton of programming, you will never get a chance to show it, because most companies will not call a math PhD back on the off chance they can do something useful. Say what you will about academia, but if you apply to an academic research group whose work has some mathematical flavor, they are much more likely to actually give you a chance.
Someone linked a talk by Cathy O'Neil in another answer. It contains good advice, but recall that this was a talk given at MIT. Cathy O'Neil has a PhD from Harvard and research experience at MIT. Her first work post-academia was at D. E. Shaw. It is reasonable to assume that she does not have any experience with the obstacles that confront the average academic who wants to transition into something else. She writes: "being really [flipping] good at math is considered a superpower by the people outside. This is because you can do stuff with your math that they actually don’t know how to do, no matter how much time they spend trying." These are the words of someone who has had a very atypical experience in transitioning from academia to the private sector (granted, given the audience of the talk, it is reasonable to assume that most of the audience will be atypical also). People coming out of academia who are not coming from the absolute top schools, with the connections that often come from that, are generally not greeted as superheroes by the private sector. (I want to make clear: I'm not criticizing Cathy here, or suggesting that she hasn't worked hard to get where she is. It's just a lot harder to do what she has done than you might think, reading only her words and not considering the context.)
I would say: if you want to get a mathy job, unless you have a documented history of things that are of immediate relevance to the private sector, or professionally useful personal connections in the outside world--- stay in academia, but switch fields to an applied area (lots of good suggestions have been given here). After a few years, you may have connections that can help you transition to a mathy job outside of academia, or at least a broader resume that people might be more inclined to take seriously. I don't mean to be negative here--- you probably should be taken seriously as you are now. But outside of academia, in my view, the odds are that you won't.
