In 5 seasons of one sticking (well over 500 climbs) I've knocked my stick off the tree 3 times now during the climb and one time had a mechanical failure of a rappel device. All four times I had to perform some means of "self rescue", luckily always successfully. The key is to not panic in the moment and to fully evaluate your options. Luckily most of us already have some means for self rescue already on us if we take the time to consider them. I'll revisit each scenario here and explain what I did and what other options were available that I could have chosen from.
First Scenario: I was on an observation sit in the evening about 15 ft up. When it came time to get down I hooked up my rappel rope and got set to rappel down. The issue occurred when I pulled the release handle of the rappel device and it came off in my hand leaving me with no apparent way to release the device.
What I did: I simply transitioned back to my stick, staying on the rappel line and one sticked my way back down the tree. Not my preferred method of descent but it's very doable and I use the method occasionally throughout the season just to keep in practice.
Options: At that time I was carrying an ATC and short prussik as a back up and I could have swapped to it and completed the rappel. I've since removed the ATC from my kit but I could just as easily swapped over from the descent device to a carabiner with a munter hitch and rappelled using that.
Stick Drop Scenarios: I'll cover these as a group since the self rescue options are similar and plentiful enough that I've used a different one each time.
First Occurrence: Morning hunt climbing in the dark. I was on my second stick move when I think I pinched the tag end of my attachment rope under my foot stepping into the aider. At any rate when I stepped up the stick fell off the tree to the ground. Because it was dark and I couldn't make out the stick on the ground below I took my rappel rope from the pouch of the turkey vest I use as my pack and girthed it to the tree and hooked the rappel device into my bridge. I was able to release the tether friction hitch transferring my weight to my rappel device and then I simply rappelled down. I then retrieved the stick and restarted my climb using the rappel rope this time since I was already on it.
Second Occurrence: Evening hunt, I felt I was late getting into my spot and selecting a tree so I was hurrying on my climb. I was one stick up and I think I didn't get the stick attachment rope fully into the cam cleat and failed to "set" the stick prior to stepping into the aider. Once again the stick dropped off the tree to the ground. This time it was daylight and I could readily see the stick so I broke out the grapple and string I carry to retrieve stuff I've dropped while hunting and simply grappled up the stick while hanging in the saddle. Once it was reattached to the tree I simply continued my climb.
Third Occurrence: At this years Michigan saddle hunter's get together I was trying to demonstrate a setup to another attendee and tried to rotate my stick on the tree while I was hanging in the saddle. The rope went slack and the stick dropped to the ground leaving me suspended from the saddle about 15 feet up. I first attempted the transfer to my rappel rope as above but the friction hitch I use on my tether was set tight enough that I struggled to break it loose to transfer my weight to the rappel rope (FWIW, I'm confident I could have grabbed the hitch with both hands and pulled down on it and gotten it to release enough to transfer over but it wasn't the easiest option available at the time). My next step was to pull the platform from the pouch of my vest and attach it to the tree at waist level. From there I was able to climb up on the platform and unweight my hitch and complete the transfer to the rappel rope.
Unused Options: There are a couple of more options I can think of that would work for me in the "stranding hanging" scenario that I haven't used yet. The first, would be to take my linemans rope out and use it as a foot tether. That would allow me to unweight my climbing hitch enough to make the successful transfer to the rappel device.
The second option would be to simply use that "second tether" and 2TC down the tree. I'm not at all versed in the 2TC method but I'm confident I could inch my way down the tree using it if I had too.