The use(s) of of with quantifiers
The usual function of "a couple" can be described by the label of "quantifier".
Other words that can act as quantifiers include numerals, more, and all.
Many quantifiers cannot directly precede a definite noun phrase: we can't say "twenty the children" or "more the children". Since a pronoun like them stands in for a definite noun phrase like the children, I don't think it should be too suprising that we also can't say "twenty them" or "more them". Instead, the quantifier has to be followed by a prepositional phrase headed by of: "twenty of the children, more of the children, twenty of them, more of them". In this situation, the quantifier has a partitive sense: these expressions refer to only part of the children, not all of them.
Less straightforwardly, there are quantifiers like all that can be used directly before the definite article ("all the children") but not directly before a pronoun (*"all them"). Of is mandatory in all of them and optional in all of the children (there are some prescriptivist arguments about it here).
So the reason for the position of of in "a couple more of them" is pretty straightforward and unrelated to the presence of "a couple": *"more them" is ungrammatical.
The "adverbial" function of some quantifiers
Garner isn't alone in calling some uses of quantifiers adverbial, although I'm unsure of the correctness of this analysis. Another quantifier that is often described as being used like an adverb is much in the expression "much more" (also many in "many more").
There are similar examples of quantifiers that can be used "adverbially" in some other Indo-European languages, such as French (tout/toute/toutes) and Spanish (mucho/mucha).
These words display anomalous morphology for either either adverbs or determiners in that they show (different kinds of) partial agreement patterns: much is ungrammatical in English directly before a plural noun (*"much coins") but acceptable for some speakers before fewer followed by a plural noun ("much fewer coins"; alongside a variant "many fewer coins"); adverbial tout agrees in gender but not in number (aside from a purely orthographic convention for the the feminine form, which is pronounced as /tut/ but spelled as either, toute, toutes or tout depending on the context), and adverbial mucho has an optional feminine singular form mucha in some contexts but is invariable mucho in the plural in both genders.
Some analyses use terms like "predeterminer" to describe quantifiers that come before a determiner, but I'm not sure how rigorously defined the concept is.
Dialectal variation
If all English speakers had the same usage in this area, Garner probably wouldn't have felt the need to give the advice that
There is no place in the example for of (neither ✳a couple of more shrimp nor ✳a couple more of shrimp makes sense).
In fact, phrasings like "a couple of more days" do seem to exist, but it is less common than "a couple more days" without of. Here's a Google Ngram chart:
To many speakers, including me and apparently Garner, the "a couple of more" construction doesn't sound acceptable.
