Not for multiword phrases, no.
an attractive eighteenth-century solid-gold necklace
Sometimes you can hyphenate if you must, like if there's some ambiguity otherwise, but it usually looks better without that.
an old sugar maple tree stump
The only adjective there is old. But when you have never learned that there can be anything else, you get random painful mistakes.
> In the teaching of high school English, the terms 'adjective' and 'modifier' seem to be used relatively interchangeably. From the point of view of linguistic theory, this usage reflects a lamentable lack of appreciation for the central linguistic distinction between function and form.
> Modifiers are linguistic expressions that serve a certain function---namely, to restrict or qualify some other expression. Adjectives, on the other other hand, are members of a syntactic category that is defined by certain formal properties. For instance, it is possible to derive adverbs from many adjectives (heavy, heavily; mere, merely; rough, roughly; sweet, sweetly). Like any other syntactic category, adjectives project intermediate and maximal projections (= adjective phrases, AP, AdjP).
From here. Save that link, you'll want it later. I promise.
There's more to be read there. It's good stuff.
> There is no one-to-one relation between modifiers and adjective phrases. Modifiers are not necessarily adjective phrases, as illustrated in (1).
(1) a. Adjective phrase a very aggressive driver
b. Adverb phrase They drive very aggressively.
c. Prepositional phrase the desk next to the window, they arrived on time
d. Noun phrase They will arrive Monday night
Conversely, adjective phrases are not necessarily modifiers. For instance, the adjective phrase in (2) is the predicate of a small clause.
That's college level English in this country.
High school and earlier never get it right.
And nobody takes English grammar in college.
Except linguistics students.
So it's very, very frustrating to answer questions correctly in the face of the pushback by folks convinced they're right because that's what they learned in the fifth grade.
"syntactic category that is defined by certain formal properties" is not something ANYBODY shy of university linguistics students is ever, ever taught.
You can't go back to the grade-school analyses. They break down horribly.
> Not everything that functions as modifier in clause structure is an adverb; though some grammars and dictionaries take that view, it falls apart under close analysis and serious grammarians have mostly abandoned it.
Wrote DW256 in his ELL answer.
There's so much schlock on the internet you can always find something daft as a doorknob supporting your view. And that seems to justify being obdurate about it.
He's right, and it's exhausting to have to retrain everybody who never got past fifth-grade levels of grammatical analysis.