From what I can see the time between signing up & actually shipping out is about 5 months. That all depends on the MOS you choose. Just tell your Recruiter that you can't leave before a certain date & he'll make sure that happens.
You still have to meet with a Recruiter, go through the recruitment process, take your ASVAB, get your physical & meet with the MEPS Counselor to chose your MOS & that all takes time, certainly more than a month in most cases.
To be doubly sure then you should make an appointment with a Recruiter after 2 October. That's only about 3 weeks away.
I should add that you can get the ball rolling with a Recruiter right away. I don't feel there's any great rush for them to get you in right away. If someting DOES come up where a particular MOS has an immediate opening then just turn it down explaining why.
It just doesn't go that fast normally. And actually you can reserve an MOS & stay in the Delayed Entry Program for up to a year before shipping.
Now you have some time to study for the ASVAB & getting yourself in shape - at least good enough to pass the PT test before you leave for BCT.