Doing a reenlistment jump is something I'm offering everybody I work with. I only had one person take me up on the offer so far and he had the time of his life. As long as you have Old Glory in the plane, you can do whatever you like.
After the oath, we took the flag and folded it. Then I put it down my shirt, jumped with it, and then presented it to him on the ground.
Here are some pics...
Clicky