How to transfer from a pure math PhD to an applied math PhD at another school?
Get your feet wet in applied math to make sure you'll like it.  Apparently there is no applied math program where you are now, but you can still do this,  through a combination of self study and courses offered in your own and in other departments.  Look at some applied math programs at other universities to get a flavor for the variety involved, and to give you some ideas of what sorts of courses to look for.  Programming skills are not required for all areas of applied math but can really come in handy for some.
When you're ready to ask for recommendations, be honest, just say that you are hopeful an applied math degree will make you more employable.
I assume you'll want to pursue this process with some transition time, during which you'll continue working to some extent with your current program including your research project....  If so, please find some people to interact with remotely, in your area.