It sounds like you have no idea what the career options for a Math PhD in Australia are like, which is understandable but not a good place to be. It's important to educate yourself about what the market is like, preferably as soon as possible. Since you're in the US, you won't be able to rely on your university's career center, so the burden of research is on you.
Go to an Australian jobs portal such as SEEK. Search for jobs requiring a PhD in math. If you know where you will settle in Australia already, then narrow only to jobs in that city. You'll get results like this one.
Our client is a leading provider of form and data software solutions for the global racing industry. They are an established and stable organisation who are perfect for those sick of the corporate life.
Working with some of the best mathematics and stats minds in the city in a small and focused team environment, you will work on challenging and rewarding modelling projects.
To be considered for this role you will have the following:
- A strong background in mathematical and statistical modelling
- Previous experience working as a Quant Analyst, Data Analyst, Data Scientist or Big Data Analyst - desirable
- Masters or PhD in maths / statistics / machine learning / econometrics essential.
- Experience using C++ or Python is highly regarded, although Matlab & R experience will also be considered.
- Strong algorithmic coding skill and solid knowledge of data structures
- Application clustering / Big Data experience using Apache Spark
Experience with statistical forecasting essential.
- Understanding of machine / statistical learning techniques such as non-linear regression, kernel regression, support vector machines (SVM), neural networks, classification trees and similar techniques very beneficial.
- Experience working within the horse racing / wagering industry highly desirable
It's up to you to decide if you find this job attractive. If the answer is no, then search for another job until you find something attractive. Once you have an attractive job, do your best to pick up the necessary skills. For example to be competitive for this job, you really want C++ or Python proficiency, and preferably Matlab and R as well. You want modeling experience, you want to understand neural networks, classification trees, etc.
Your PhD is an opportunity to pick up these skills. As you say, right now you have very little programming experience which immediately rules you out of this job. If you still want to do it, then should you get the chance to direct the trajectory of your PhD - e.g. if your supervisor asks you "what do you want to do next" - this is your chance to learn those skills.
As for whether it's better to do a second Masters degree, well, you can apply the same methodology. Search SEEK for jobs requiring a Masters in computer science or machine learning, and see if you find those jobs attractive (relative to the ones that need a PhD in math). If the answer is yes, then it's probably better to do a second Masters.