I looked into a public health master's options when considering which MSN to choose. I wasn't looking at quite the same thing, since I already had a BSN, but this is what I found.
You've basically got three options. MPH, MSN/MPH dual degree, and MSN with a pubic health emphasis.
You should be able to be admitted to an MPH program based on your previous bachelor's. MPH is the industry standard for public health, and programs are widely available. Ideally, you want a program that is CEPH accredited - it may no matter if you're planning on working only within your local community, but it will maximize your options. MPH will generally not take the place of a BSN for jobs that require BSN. I'd imagine most local public health departments would consider ASN + unrelated Bachelor's + MPH perfectly sufficient. However, if the job you're interested in specifically requires a BSN, it might be a good idea to ask first.
In MSN/MPH programs, you earn both degrees. This is, from what I've seen, the most time/work/money intensive way of doing it, since there is some overlap in curriculum, but you're still doing two different master's programs. This would give you the most employment option, since you actually have both degrees. I do not know if there are any RN to MSN/MPH programs - if the university offer both RN-to-MSN and dual MSN/MPH, it would likely be an option.
There are some MSN programs with a Public Health emphasis. These won't qualify you for jobs requiring an MPH - it's just a quirky MSN emphasis (I say that as someone who came close to enrolling in such a program before discovering something that was a better fit, not as an insult). For what you want to do, it would probably be fine. There aren't a whole lot of these programs, but they're out there. Grand Canyon University and UNC Charlotte have RN to MSN-PH, and if those two exist, there are probably others.
Both MSN options would take the place of a BSN as far as employment goes.
Also consider that there are relatively quick and cheap RN to BSN options, and doing that would expand your program choices, since you wouldn't be limited to those that have bridge programs.
If your goal is teaching, an MSN Education program might be a better fit, and RN to MSN Ed programs are very very common. Most local public health departments are not going to require a Master's degree for this sort of position, let alone something specifically public health focused.
I've seen 100% online options for all of these.