In particular, the ex- befuddles me, because synekdekhesthai doesn't appear to signify any notion of outness or outwardness! I quote Etymonline.
"figure of speech in which a part is taken for the whole or vice versa," late 15c. correction of synodoches (late 14c.),
from Medieval Latin synodoche, alteration of Late Latin synecdoche,
from Greek synekdokhe "the putting of a whole for a part; an understanding one with another," literally "a receiving together or jointly,"
from synekdekhesthai "supply a thought or word; take with something else, join in receiving,"
from syn- "with" (see syn- + ek "out" (see ex-) + dekhesthai "to receive," related to dokein "seem good" (from PIE root dek- "to take, accept").
Typically an attribute or adjunct substituted for the thing meant ("head" for "cattle," "hands" for "workmen," "wheels" for "automobile," etc.). Compare metonymy.