「もっと」 certainly can modify nouns, but not the nouns used in your examples like 「本{ほん}」、「対策{たいさく}」 or 「お茶{ちゃ}」. You are indeed letting the English translations get in your way of understanding Japanese on a deeper level. (You are not the only one who does this, I assure you. I myself did the same when studying English.)
While "Read more books." may be a valid translation of 「もっと本を読みなさい。」, it is not what the Japanese sentence means to us Japanese-speakers. It means "Read books more." to us. In other words, 「もっと」 modifies the verb 「読みなさい」, not the noun 「本」.
The exact same thing can be said about your other sentences. What word is modifying what word can and should only be determined from the Japanese original, not from the English translations.
"Think of countermeasures more." > "Think of more countermeasures."
"There are books more." > "There are more books"
, etc.
Which one of the pair sounds better and/or more natural in English is completely irrelevant in the discussion of Japanese grammar and vice versa. Some things are always lost in translation.
What nouns can 「もっと」 modify, then? Those are the nouns that can be discussed in terms of degrees, not in terms of quantities. Those nouns include 「右{みぎ}」、「左{ひだり}」、「上{うえ}」、「下{した}」、any one of 「東西南北」, etc. 「もっと右」 = "further right", 「もっと南」 = "further south", etc.
Hope you can see the difference between those nouns above and nouns such as 「本」、「お茶」, etc.
Can もっと mean 'additional'?
No, it can't for the reasons I just explained.
「ここにはもっと本がある。」
"There are more books here (compared to elsewhere)." is just someone's translation. (Yours?) In the Japanese original, 「もっと」 modifies 「ある」, not 「本」.
Finally, a brief talk on what other things 「もっと」 can modify...
・ i-adjectives: 「もっとかわいい子」 ("a cuter girl")
・ na-adjectives: 「もっときれいな子」 ("a prettier girl")
・ verbs: 「もっと食{た}べたい」 ("I want to eat more.")
・ adverbs: 「もっとしっかり勉強{べんきょう}しなさい。」 ("Study harder!")
「もっとお茶はいかがですか。」
In this sentence, 「もっと」 modifies 「いかが」. It can because 「いかが」 is an adverb.
