Don't ordinarily hyphenate adjectival combinations of adverb + adjective or adverb + participle unless the adverb does not end in ly and can be misread as an adjective.
Now--if that sounds like just a bunch of silly grammarese to you, let us put it simply: Don't hyphenate stuff like "fully involved" or "partially hidden treasure" -- if you do, people in the know who read your ad, proposal, white paper, cover letter, or contract will move it to the bottom of the stack and give preference to your competitors who make fewer mistakes. That's just a fact. Read on and find out if you're making other similar mistakes.
Since the function of adverbs is to modify adjectives and verbs, and since participles are merely forms of verbs that can act as adjectives, the combination of adverb + adjective or adverb + participle is just a simple case of one word modifying another, and ordinarily no hyphen is needed to show the relationship.
An appropriately-red bridal gown and a completely-confused groom are errors in American English; there should be no hyphens. (The British often do hyphenate such compounds, however.) Some writers are misled by three-part compounds, such as a badly run-down neighborhood, and insert a superfluous hyphen after the adverb: a badly-run-down neighborhood. In this example there should be no hyphen between badly and run-down (which is correctly hyphenated as a participle + adverb adjectival combination).
Note that a scholarly-looking water skier is not an error. Scholarly, leisurely, and a few other adjectives end in ly, which is the standard ending for adverbs, but they are still adjectives, and the combination of adjective + participle, as in scholarly-looking, should be hyphenated.
Adverbs that do not end in ly and can be mistaken for adjectives
An ill-clothed baby is not an error, even though ill is an adverb and the combination is adverb + participle. The reason for the hyphen is that ill can be misread as an adjective, meaning sick and directly modifying baby rather than the participle clothed. The hyphen links ill to clothed.
It is rare that the omission of such a linking hyphen causes real ambiguity. Even so, we naturally avert confusion in speech—we almost invariably run together such combinations as ill-clothed when they precede the word they modify and often when they follow the word as well, but we are likely to pronounce combinations with ly adverbs, such as badly clothed, as two distinct words. We should do the same in writing, running together certain combinations with a hyphen. Sometimes the ear is the best judge when a hyphen is desirable, but there are some general principles and also some common conventions for specific words used in compounds.
There are many adverbs that do not end in ly and can also be adjectives, among them half, well, better, best, fast, slow, little, and long. The eight listed and some others should routinely be followed by a hyphen when they are used in adverb + participle compounds that come before the modified word: half-asleep audience, well-dressed student, better-clothed editor, best-written book, fast-moving boats, slow-moving traffic, little-used brain, long-awaited retirement.
Not all these compounds need hyphens when they follow the modified word, but some do: The parent was well dressed, The editor was better clothed, His brain was little used, and The speech was long awaited; but The audience was half-asleep, The traffic was fast-moving, The van was slow-moving, and probably The book was best-written, though the last example is an odd one that would be unlikely to occur.
Usage varies on fine points of punctuation, though, and the ear is often the best judge. Half is particularly variable. In one of the examples in the preceding paragraph, The audience was half-asleep, the hyphen seems desirable, perhaps because without the hyphen one might think the sentence meant that half those in the audience were asleep and half were awake. But often it does not: The man was half dead, The door was half open, The meal was half finished. It is also tricky when it is used to modify verbs, usually requiring a hyphen before transitive verbs but not before intransitive verbs: He half-turned the knob, but He half turned and looked out the window.
If you're having fun with all this by now, you may be a true grammarian. If you are half-asleep, however, all this may be beneath you and you would just as soon turn your rocket-science white paper over to some really good technicians of words to make it perfectly acceptable to that famous journal you're aiming for. We understand.