I agree with your current suggestions because between your experiences and research I can't tell if you understand the demands of an Epi program or why you would want to pursue a PhD in Epi specifically.
Coursework: A more relevant masters would be good. Perhaps a more affordable program such as online/blend since you already have a masters. Instead of a master, you could also take a few classes or do a summer workshop. Here is an example, hosted by Columbia University: https://www.mailman.columbia.edu/research/epidemiology-and-population-health-summer-institute-columbia-university-epic
Research: Additionally experience working with data sets and using statistical software(s) - authorship would be nice but not necessary. You should build up your skillset related to epi. What exactly was your sample size, saying "large" is vague.. was it over 300? Have you considered trying to get the manuscript published?
Experience: I'm not familiar with programs you can apply to; however, applying to them would only strengthen your application if this round doesn't end well. Also, look at the publically available resume/CV of people working in the epi field - see what work/programs they completed before entering their doctoral program.
Hope that helps! A little about me - I submitted my apps for Fall 2019 for Behavioral Science/Health Ed program. I've been working in research of 8 years including my undergrad years. I have an MPH. I've been working as a public health researcher at a med school full time for 3 years as the lead on several projects.
Good Luck!