The most common situation where an adjective following a verb should not be changed into an adverb is when the adjective is the subject complement of a linking verb. You can tell this case because such adjectives are all complements of some form of the verb "to be" or could be preceded by the phrase "as if" or an equivalent without substantially changing the meaning. In essence, such verbs are indicating some form of identity and so are incompatible with adverbs indicating manner.
An extension of this usage occurs when certain adjectives describe the state of the subject as it performs the action of the verb, the state the subject ends up in, or the state the object is in at the start or end of the action. You can identify this case, be seeing if you can precede the adjective with "(while) being...."
The phrase "the layer of haze starts out very tenuous" is an example of this extended usage. It is a reduced version of "the layer of haze starts out (while) being very tenuous."
A native speaker might also venture to say: "the layer starts out tenuously," this presupposes the semantics of a "tenuous start," which is also possible, but less idiomatic for a non-volitional subject.
Contrasting examples that clearly distinguishes the need for this extended usage are the following two sentences:
He angrily walked in.
He walked in angry.
The first sentence can suggest that the person showed visible signs of anger as he walked. It would be the equivalent of "he walked in in an angry manner." Depending on the situation, it could also be that the person was angry at having to walk in. It would then be the equivalent of "he walked in and was angry at having to do so." In either case, there is a link between the anger and the walking.
The second sentence merely describes the mental state of the person, contrasting it with other possible mental states. It is a reduced version of "he walked in while being angry." The only link between the anger and the walking is that the state of anger existed as the walking proceeded.
A common public announcement in the US is "Don't drive drunk," which means don't drive while being drunk." It does not really describe how the driving looks. If you said: "Don't drive drunkenly," it would mean "don't drive in the manner of a drunken person" and doesn't describe whether the driver has drunk any alcohol.