Horace, if you're right about the hood, perhaps there is a tie-in with the Nine of Pentacles card where there is a hooded falcon (and often a reference to the woman having her thoughts/instincts under control). If this is the case, it is interesting that the head is on the bag, which is where his future "tools" to help him on his spiritual journey are hidden.
Fulgour, thanks for sharing the information about Le Bateleur and the hawk. Having seen many hawks in my neck of the woods drop like a stone from the sky (going after some lunch), it does indeed make me think of the "dive" the Fool is about to make from the cliff!
Bodhran