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If you want a job as a software engineer you will need to demonstrate that you can write software, regardless of your education. So I would recommend (as others have mentioned) that you put your work out there on github or (even better IMO) contribute to an open source project. Working on code with others can provide a tremendous education in itself.

There are a lot of people who will see your PhD as a black mark when it comes to software engineering. There is a belief that you are over-qualified (what ever that is supposed to mean). Whether it is a reasonable belief or not it exists so you will being trying to prove that you produce Quality Code and a Team Player (TM). Once you get your foot in the door, it should become easier.

There are also lots of places that are math PhD friendly :) Like the mathworks for example!

FTR, I have a PhD in applied math and have been a software engineer for about 8 years now. I got my foot in the door by working for free at a game company for a year. Not the way that I advise everyone to take ;) but I lucked out and learned a lot from the experience that it continues to influence my view on software, teams, and leadership to this day.

Good luck!

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