I've noticed that he's using his thumb to mute, so he's probably not holding his pick in between his index and thumb, which would suggest it's in his palm. Does anyone know this technique, and can explain it? He seems to be able to take his pick out quite quickly.
As @RockinCowboy pointed out, for most of the tapping sequences he simply tucks the pick into his palm and holds it there with his 3rd finger. Retrieval is simple, as you can see at 0:40, in fact at around 3:10 you can see very clearly how he just uses his 3rd finger.
There is a sequence around 1:35 where he holds the pick normally and taps using his 3rd finger - but this looks to be because of the sequence immediately prior.
(As an aside - he isn't actually muting with his right thumb. For most of this track he is using it to position his tapping fingers accurately.)