A copywriter I'm working with wrote "One-Day Only Promotion" but my feeling is that "One-Day-Only Promotion" is correct. The first three words describe 'Promotion'. I know you don't hyphenate adverbs, but does that apply when one is part of a compound adjective?
It should be one-day-only promotion, for the reason you reported.
Have a look at some more genuine, authentic, certified, they-really-happened hyphos (a word we've made up on the model of typos): sung-lasses, barf-lies, warp-lanes, doork-nobes, broom-sticks, pre-gnant, air-trips, boot-traps, stars-truck, sli-pup, ong-oing. —Comma sense, Richard Lederer, John Shore
I agree that "One-Day-Only" would be correct.
But copywriting is the Niagara Falls of bad grammar, so I've grown used to poor usage there. I often see this as "One Day Only Promotion" and my hide has been thickened over time to the point where it doesn't hurt me any longer.
It is normal practice, in journalism at least, to consider the first compound noun in a compound adjective as a complete entity, so only one would be used. For example,
New York-style pizza
Nevertheless, if the compound modifier is long (and the first part is not a proper noun), it is not uncommon to hyphenate all single words in it.