Let's look at this (I have never paid any attention to material components before, beyond being vaguely aware of them as fluff while playing a spellcaster - which I don't do very often at all):
Friends "a little makeup applied to the face"
Hex "a petrified newt's eye"
Mage Armour "a piece of cured leather"
Minor Illusion "a bit of fleece"
Unless you think Burton has been following the prison rules to the letter, I think he could have squirrelled all of that about his person (say, sewn into a false pocket in his prison trousers). Just in case. (No pun on your name).
Of course, the 'makeup' is a bit of soot soaked in rationed beer, the 'petrified newt's eye' is a dried rat's eye, the 'cured leather' is a piece stolen from a worn-out shoe, and the fleece is from your bedding (you got fleece instead of straw because of your nobility, lucky you).