Can “another” be used with plural nouns provided periods or measurements don’t count?

Solution 1:

It is perfectly acceptable in any but the most formal contexts. It probably would not be used in legal or diplomatic texts, which must avoid any possible ambiguity, however far-fetched; but it's fine in anything less restricted than that. Here, for instance, is a footnote from an impeccably academic text, Jon B. Sherman, The Magician in Medieval German Literature, University of Illinois at Urbana-Champaign, 2008:

Thorndike eventually continued his work, adding another three books that investigated the post medieval period up to the seventeenth century.

And it's not a modern vulgarism, either. Here's an extract from William Burt Harlow, An Introduction to Early English Literature, 1884, p.132:

By 1595 he had completed another three books of the "Faery Queen."

(I should perhaps add that this use is not confined to threes of others—“another three” was my Google search term.)

Solution 2:

Using another in this way forms a commonly used English phrase.

Per the Macmillan Dictionary:

another two/ten/hundred etc.

used for saying how many more people or things there are

If you're confused about the number of things that can follow another, Macmillan has this usage note in the link that I cited:

Another can be used in the following ways:

as a determiner (followed by a singular countable noun): Can I have another glass of water, please?

as a pronoun (without a following noun): We're changing from one system to another. (followed by "of"): I have another of his books somewhere.

I hope this answers the questions that you raise in your comments.

Solution 3:

I agree with OP - then another one sounds "normal". It's almost as common as then one more...

...whereas then another two doesn't sit so well. On average, we definitely avoid it...

I also believe OP is correct when he says that a span of time, for example, can be treated as a "single unit" in this context. If you say "I'll be ready in another five minutes", you're thinking of a period that long, not "five times the duration of one minute".

Google Books reports 2590 hits for then another five minutes, but only 192 for then five more minutes, so obviously that strong bias against another+plural doesn't apply to familiar time-spans.

I don't think there's any real "point of grammar" involved - here's Professor Michael Swan at the BBC World Service...

There’s one odd thing about another. You can use it before a plural expression with a number.

In short, whilst there's nothing grammatically preventing us asking the greengrocer for "another three apples", it turns out we're much more likely to ask for "three more apples". I think that's partly because we don't normally think of three apples as a unit, the way we do five minutes.