I've been using the "gift wrap" position since before it was called the "gift wrap". I learned it as the "headlock mount".
Most of the chokes I learned from this position I learned way back when I was a white belt in like, '97, and over the years I've been kinda surprised how many guys don't know this position that well or how to attack from it. It is brutal in MMA when you can punch. Its practically game over.
A lot of what I've learned to do over the years is get to that position. It is easier than you think. Some of how I learned to move more efficiently to that position I picked up from watching Saulo's No Gi Revolution video. Mostly just paying attention to moving uke's elbows away from his side.
The chokes I do from there kinda depend: I can work into an arm triangle, do an old catch style "cobra clutch" (this is tricky and takes lots of practice) or do another choke that an older Brazilian black belt I know calls, "the Rickson Gracie". I showed him the choke once (since he used the giftwrap a lot to devestating effect and showed me the most wicked no hands arm lock I've ever seen and FELT from there) and he said, "Oh yeah, I know that one, like Rickson Gracie does."
Funny thing is my first instructor whom I learned the choke from was originally a Rickson Gracie student. The choke is one of my favorites and is one of the meanest.
Also I can attack both the near arm and far arm from the gift wrap position. Or I scoop the far arm from the gift wrap with a deep underhook while maintaining the giftwrap and take the back. You have such good upper body control that you can kind of "steer" them into your rear mount.
