The Chicago Manual of Style, sixteenth edition (2010) glancingly affirms the advice that Barrie England cites from The Penguin Guide to Punctuation in his answer:
6.80 En dashes with compound adjectives. ...
A single word or prefix should be joined to a hyphenated compound by another hyphen rather than an en dash; if the result is awkward, reword.
a two-thirds-full cup (or, better, a cup that is to-thirds full)
As Chicago very wisely observes, the critical question to ask in deciding whether to use multiple hyphens in a compound word is a highly subjective one: Does the result seem awkward? If it does, your best bet is to reword to avoid the awkwardness—not to press on in unthinking obedience to some set-in-stone rule about proper punctuation, regardless of the effect it has on the sentence you're trying to construct.