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Guys will get picked up to Test Pilot at the very end of their JO tour, or very early at their first Shore Tour depending on when the boards are and when they actually got to the fleet. There are also several ways to become Test Pilots, and there is a selection board at Millington. You must submit a package, endorsed by your CO and probably CAG and the Admiral. The package goes to the board and you compete agains a couple hundred other guys. If selected, there are a couple different ways to actually become a tester... AFIT, Direct Entry, etc... and after you graduate from TPS, you owe your payback tour as a tester at one of the VX's. Some guys have time after their test tour to have a mini-tour doing something prior to DH screen... especially if they don't screen for DH on their first look.
Unknown about how the AEDO job works, but I was under the impression that guys who were pilots and became AEDO's don't fly anymore.