There is a difference in usage and meaning between 「こ」 and 「こっ」.
While 「こ」 is generally used as diminutive, 「こっ」 is used for different meanings and nuances.
デジタル大辞泉 defines 「こっ」 as:
[Prefix] 《Geminated version of prefix 「こ（小）」》Attached to adjectives and sometimes verbs for added meanings of "slightly", "considerably", "heavily", etc. 「―ぱずかしい」「―ぴどい」「―ぱずれる」
Note that unlike 「こ」, 「こっ」 cannot be attached to nouns. (Your examples 「小雨」 and 「小柄」 are in the structure "小 + Noun": therefore, it is not a fair comparison.)
Thus, at least by dictionary definitions, there is nothing strange with what 「こっぴどい」 means, which is 「とてもひどい」.
If I were to add my own perspective here, the word 「ひどい」 contains a fairly negative subjective value judgement to begin with. In other words, it would not be very natural to say 「すこしひどい」 or the like.
In conclusion, what the prefix 「こっ」 means depends on (1) the word that follows and (2) the context and what the speaker means to express. For instance, people do say 「こっぱずかしい」 to mean both "a little embarrassed" and "very embarrassed".
Advanced learners might also need to know that 「こっ」 is sometimes used more for the good rhythm it gives to one's speech than for the added meaning/nuance. In particular, the small 「っ」 combined with a word starting in the p consonant gives the explosive kind of rhythm. The p sound is rare in the language to begin with, so we "create" it by using rendaku with words starting with a は-line syllable. The three examples above from デジタル大辞泉 should tell you something about it -- 「―ぱずかしい」「―ぴどい」「―ぱずれる」.