If you're planning on attending Grad School, then I would recommend looking into a Management Internship. Utilize the contacts you made during CP to network as MIs can be very competitive, but can move you quickly into a FT Manager position.
Each Line of Business operates their LCC differently - some require several rounds of interviews, some do not. However, they all want you to have experience within their specific area. So focus on the area you did your CP or consider going for a FT hourly position to gain the experience in the area you're interested. Once your back, demonstrate your interest in training... then coordinating. LCCs typically post once a year, so network 4-6 months before the posting and make sure your current Leaders will support you once you apply.
If you're not interested in coming back in an hourly capacity, you can always apply for Management through Disney Careers. It's not officially the LCC, but it gets you to the same ultimate goal.