i'm not familiar with this particular plant but i'll just write up what i can guess at.

take it with grains of salt.
i've only grown a few kinds of tall phlox and one of them is the wild kind which is purple, pink and white, gets plenty of seeds and spreads around wherever. the goldfinches love them, eating the seeds as they form and ripen. the other type of phlox we grew is a tall pink kind which was supposed to be red, but never actually flowered at all, we've never gotten flowers on it and after about 10 years we tried to dig it out, i'm not even sure we got it all nor do i really care.

it was not easy to dig out. and seemed to come back from bits of root if they were left behind (for all i know we were sent a weed).
so my guess is that you could divide it and i would do that in the middle of spring after the ground has thawed enough but before the main growing season has started for the plants. since i've not dug one up in ages i can't say how i would divide it, but usually a larger plant can just be cut into two, three, four, etc. parts as long as each section can have parts of the growing active budding area and some roots (you can examine plants now and into the fall to see what they look like as they grow and how they are formed, what the roots are like, etc.).
i'm generally pretty fearless when it comes down to dividing plants and have reasonably good results - but it is funny and coincidental that the plant that i've had the hardest time with propagating here is some creeping pink phlox. it just does not like clay at all and where i tried to move it was just too heavy and too dry for too long so it did not continue and in the moving i killed off quite a bit of the creeping phlox and i was really hoping to get it to grow. luckily the creeping thyme i've since put in has done the job and it doesn't mind the rock hard clay at all. it may take longer to grow to cover the clay, but it won't die off.