Your background is highly attractive, and there are enough people here with the clout to get you started on a more positive interview track.

Also perhaps some coaching; you might have some tactical errors in how you approach.

With that said, finance may also be a really good option for you. At the right bank, the strong math background + top GPA is the hiring requirement. They also like people with Ph.D.s in technical physics.

You should have some coding experience, but you'll be working in their custom setups anyway -- if you're good at math, you can pick up APL (it's a language, look it up) programming quickly. Otherwise, bank python is pretty divergent from python, or you are looking at Matlab/R.

They'll want someone who can translate complicated math into a model. You are less likely to find that in most tech companies.

I agree that I may be making tactical errors. I've confined myself to job adverts through e.g. LinkedIn for the lack of knowing where else to look. I also lack the connections that most others at my stage of career have.

I know my way a bit around coding though I'm entirely self-taught so I have no degrees to back up that claim. I have considered throwing my projects on Github as a kind of 'proof of work' to employers.

Could you perhaps name some specific directions that would be suitable for someone with my background? People around me mentioned quantitative finance as an example of a direction for math PhDs to go into --- but I'd frankly be open to any job that lets me make use of my skills.

Are you good at math competitions? Competitive programming is basically the same thing. This is how I transitioned in the field. If you can do these, interviews will be a breeze.

I like to think that I could get through interviews quite easily too; if applications start with a cognitive test I invariably make it through. My applications tend to die at the HR layer (which is usually the first stage) so I haven't actually made it to a real-life interview.