Master's necessary post-PhD for switching field (neuroscience -> bio/statistics)?
Future learning may be necessary, but a formal degree probably is not. What you need is to have a  position that gives you some flexibility in what you do and a base from which to do it. Your current degree is technically sufficient, if you can find the position.
From there you can learn what is necessary, start research in the new field and put together some papers.
But, in the short term you may be required to also produce in your current field to satisfy an employer and keep your base secure. This implies it may be necessary to morph over time, rather than to make a clean break.
It might even be that if you have a position in a large university that you can sit in on the occasional course in the new field to help you fill holes in your education that you might have. Even discussions with faculty in the other field can point you to a quick path to pick up what you need.