I think sumelic is correct, though both De Vaan and Etymonline disagree. If I may quote Lewis and Short:
maxilla , ae, f. dim. of a ground-form magsula (whence māla; root mag of μάσσω, to knead; μαγεύς. baker, etc.; cf. axilla, āla, from ago)
This makes sense, too, because -xilla is not a proper diminutive suffux. So here, the root is mag- with a sibilant for the ending. Making that diminutive gives us magsula or with a more modern orthography maxilla (cf. older Latin optumus and standard optimus).
De Vaan derives it back to smaksla- (cf. OIr. smech 'chin', Hit. zamakur 'beard', Lith. smakras, 'chin', etc.). However, he doesn't really account for the loss of the gutteral in māla; instead, he derives both from PIE.
Māla reflects *smaks(V)la, whereas maxilla can be from *smakslela, or has the productive suffix -illa.
However, if we were to use other Latin words for analogy, as Lewis and Short points out, āla is impossible to derive from ag-, and it makes more sense to go this way:
ag- > axis > axilla > āla
It should be noted that x easily slips away from words, as it turned into /h/ in Spanish, s or ss in French, and s, ss, or sci in Italian.