Breaking into Quant work as math PhD. What are my options?

I feel like every year some post like this pops up. Guess it's my turn now. 

Background: Ugrad from top state school in math, doing PhD at low/mid state school. Coursework involves measure theoretic probability, (real)analysis, diff/alg geo, diff/alg top, horological algebra, ring theory, ML, already have some software package written out and downloadable. Research is focused on applied/computational topology 

A postdoc seems like it could be achieved, but I don't wanna commit to an academic route with less payoff. So I want to focus on having a desirable skillet for industry. 

For quant trading, I believe I have the desired skill set when applying (maybe not the academic prestige LOL). But for quant research, my glaring weakness is in statistics.

Does my background make me a competitive candidate for any firm that does quant trading/research or am I just perma-mogged by the T10 ugrads that get on campus interviews? Is there anything I should focus on upskilling besides improving stat knowledge and interview prep? What's something that I'm not considering? Is there a path forward?

Appreciate any comments or feedback on this would be great. Also welcome any flame for my own personal ignorance. Thanks!