In the US, long the main sources of employment that use math are efforts in US national security.
So, at one of the top dozen or so US research universities, about 60% of the annual budget is from research grants from the US Federal government. The main sources are NSF and NIH. The main fields are the STEM fields for US national security and biomedical for medicine.
So, an academic publish or perish job should try to be a job that gets grants from NSF, etc. In particular, such a job is not just for 'ivory tower', do what you want, 'generalized abstract nonsense' but is for what the NSF, etc. want to fund.
Associated with such funded academic groups can be research positions that just do research and don't teach.
Okay, here is a general view of such work: Generally the most respected work in a field 'mathematizes' the field. So, fields that got so 'mathematized' include physics, chemistry, electrical engineering, electronic engineering, mechanical engineering, chemical engineering, aeronautical engineering, computer science, operations research, and finance.
Of course for finance, see the career of James Simons.
Now being 'mathematized' is much of genetics aimed at cancer research. There see the work of mathematician Eric Lander and, in particular, his lecture
http://www.princeton.edu/WebMedia/flash/lectures/20100419_pu...
April 19, 2010, Eric Lander: "Secrets of the Human Genome".
Notice that it's possible to get a Ph.D. in 'applied math' without setting foot in a pure math department, especially if you already have a good Master's in math. So, get your Ph.D. in some part of 'engineering'. E.g., some departments of electronic engineering have profs interested in whatever in principle electronics might be used for and, thus, e.g., pursue stochastic integration and nonlinear filtering and far out directions in 'information sciences'. See the work of D. Bertsekas at MIT -- he's a terrific applied mathematician, but I have no idea what department he's in. At Princeton there is a chemical engineering prof who, really, does nonlinear optimization for petrochemical plants -- he is popular in Houston. At Stanford notice the career of D. Luenberger -- again, a terrific applied mathematician but maybe not in a pure math department. Similarly notice E. Cinlar at Princeton, a terrific mathematician but not in a math department. Also notice the career of H. Kushner at Brown's Division of Applied Math -- he worked on a huge range of topics, sometimes even nurse scheduling. Your Ph.D. might be in 'systems analysis' or 'engineering/economic systems' or some such.
Notice the broad field of stochastic optimal control, e.g., from E. Dynkin, D. Bertsekas, and R. Rockafellar. Then notice that Intel has stated that they already know how to build processors with 1000 cores and have the cores work together effectively and notice the extreme ability of stochastic optimal control to make use of millions of threads of conceptually simple parallelism.
Another field, I once did some work in, is multivariate, distribution free statistics, especially where we can assume a lot of data. There's more to do in that direction.
Then notice that with a Ph.D. you could get a job as a prof in a B school. They might need people to teach mathematical finance, operations research, statistics, 'supply chain optimization', 'constraint logic programming', 'production scheduling', etc.
Also consider mathematizing sociology, and for that notice the work of James Coleman and Pete Rossi, both past presidents of the American Sociological Association, and also Leo Goodman.
Don't miss the work on medical research by Persi Diaconis at Stanford, e.g.,
http://videolectures.net/nips09_diaconis_bamc/
along with mathematician David Freedman, e.g., as in
David Freedman, 'Brownian Motion and Diffusion', ISBN 0-387-90805-6, Springer-Verlag, New York, 1983.
No doubt the economists continue to struggle with math; consider moving into economics where your math background could be a big advantage. Considering that obviously the economists don't have even a weak little hollow hint of a tiny clue about either how to avoid a depression or how to get out of one, there are some good reasons to do good work in economics if that is possible.
If you have some ideas about 'applied math', e.g., maybe something reasonably respectable mathematically in 'data mining', 'machine learning', 'artificial intelligence', or, really, nearly anything in engineering or beyond, some math departments will be interested. Why? Because they want students and some of the glory of hot fields and applicable work. If you can also get an NSF grant, so much the better.
Likely you can do applied math in a computer science department. E.g., somewhere in California is a prof who specializes in speculative applications of convex sets and functions. You may have noticed that some computer science departments are eager to be more mathematical but are struggling with math, sometimes, even at famous departments, struggling poorly with what is commonly taught in a junior level course in abstract algebra.
Outside of academics, considering the main source of money, maybe it should be no surprise that work in US national security has long been a good place to find a job in applied math. So, consider the various military labs around DC and, then, the companies who work with them. So consider the whole universe of US national security work. E.g., computer security is a hot topic, and monitoring is basically statistical hypothesis tests with false positives and false negatives. There's plenty of good, new applied math to be done there.
Next, understand that generally there is a problem asking that someone else create the job you really want. Indeed, if you get very far from academics, you will have a tough time finding anyone with money enough to hire and also even able to write a good job description for an applied mathematician. And they would be terrified to have someone reporting to them doing work they didn't understand.
So, really, for the job you want, you need to create it for yourself. For this, the first consideration is, where will the money come from? In particular, who will your customers be? So, find something to do where customers are willing to pay for something better, use some math to provide something better, and deliver the results via computing and the Internet.