Steve gave you the general prerequisites for the Flight Medic Course conducted at Fort Rucker. The good news for you is that isn't DIRECTLY tied to enlistment in the duty position of "Flight Medic". In fact, the 4 week Flight Medic course the Army offers is not required for performance as a 68W20F, although most "Flight Medics" do eventually attend the school if they continue on flight status.
To "enlist" as a flight medic, it's really about meeting the basic qualifications of enlisting as a 68W (Healthcare Specialist), and then having your recruiter find you an open position (para/ln) in the flight platoon of an air ambulance company. So you understand "Flight Medic" isn't an MOS. It's a skill identifyer. You are enlisting as a 68W, who essentially passes an additional Flight Physical and is then put on 'flying status'. 90% of your "Flight Medic" training is done at the unit level where you will RL (Readiness Level) progress until you can essentially be the primary "medic" on the aircraft. Frankly, this has ALOT more to do with your aircrew responsibilities than it does any medical training. Honestly, as a licensed Paramedic you're already 5 times more qualified than a 68W to perform treatment modalities.
Which leads me to the civilian aquired skills program. You can enlist (because you are a Paramedic) as an E-4, and it will also reduce the amount of time you spend in AIT at Fort Sam by 5 weeks. (This is simply skipping the Basic EMT portion of the class). You have to have your NREMT or NREMT-P certification current if you are going to use this program.
You can imagine "flight medic" positions are popular and some units require applicants to interview with the unit before handing over a position to a recruiter. This may or may not be the case with an Air Ambulance Company/Platoon near you. Your civilian experience will give you a leg up with that piece if it's required. Please feel free to contact me if you have further questions. In another life I spent 280 hours in a UH-60 airframe as a 68WF20.