Is there an hourly requirement they want? What kind of flight experience does the chief pilot and the other pilots have? All this will be very relevant when you go to apply for the job. You might have to have your commercial license for that job. Better get more info from the chief pilot as the commercial will cost you a good chunk of change.
If you go to a local flight school, meet with a few instructors and let them know your plan. Let them know you want to knock out your private in a month and see if they would be willing to do something that fast and have the schedule flexibility to do so.
There are fast track schools out there, but I never attended one so have no advice.
Getting an early start on your written is good, just remember the examiner will also be giving you an oral examination to test your knowledge applicable to the private pilots license.