When I transitioned from C25K to running a 10k, I looked up a whole bunch of 5k - 10k plans and then ended up not following them because of my schedule.
What finally worked for me was not having a training plan so much as a training formula.
I knew my goal was to run 6.2 miles without needing to take walking breaks. I was running three times a week, so I kept two runs a week at 3 miles each, and then every other week I added a mile to my third run, making it my "long run". Doing that, you would definitely be able to run 6.2 miles by the end of September.
That said, if your goal is not just completion, you can consider doing an interval workout or a hill repeat on one of your shorter runs each week.
I found it to be too much to do both a hill workout, and an interval workout, and a long run each week since I was new to running while I was working towards my first 10k, and I really just wanted to finish.
Listen to your body, and do enough to feel accomplished but not so much that you're getting injured. There's lots of training plans out there, many of them free, and many available in books you can check out from a library and look into to see trends in what the experts recommend.
Looking at training plans helped me realize that most recommend taking a day off from running after your long run rather than before your long run if you're not going to do both, and that once you're a more experienced runner, it's fine to run back to back days (this was something I had to adjust to after being used to taking a day off in between runs during C25K)
An often repeated rule is not to add more than ten percent to your mileage each week, and don't do that every week.
Going from here to a 10k in September is doable, especially if you remember that you're doubling your distance and that's enough of an accomplishment without worrying about your pace!