Additional post to provide a bit more detail.
Appearance of 高校
FWIW, it appears that the term 高校 first appears around 1949, shortly after the 1948 promulgation of a new Japanese educational law, reformulating the educational system.
Japanese abbreviation patterns
Japanese abbreviations are much more commonly derived by choosing the first component of each two-character portion of a larger compound. Thus, [世]{●}界[銀]{●}行 ("World Bank") is abbreviated as [世]{●}[銀]{●}, [下]{●}院[議]{●}員 ("member of the lower house of the Diet") is abbreviated as [下]{●}[議]{●}, [保]{●}健[体]{●}育 ("health and physical education", the subject taught in schools) is abbreviated as [保]{●}[体]{●}, etc.
We even see this pattern in borrowed terms, where the first component isn't a kanji, but instead (usually) the first two mora (kana). For example, [ワ]{●}[ー]{●}ド[プ]{●}[ロ]{●}セッサー is shortened to [ワ]{●}[ー]{●}[プ]{●}[ロ]{●}, [ス]{●}[タ]{●}ー[バ]{●}ックス is shortened to [ス]{●}[タ]{●}[バ]{●}, [コ]{●}[ス]{●}チューム[プ]{●}[レ]{●}イ is shortened to [コ]{●}[ス]{●}[プ]{●}[レ]{●}, etc.
As such, we'd expect to see [高]{●}等[学]{●}校 abbreviated to *[高]{●}[学]{●}. But instead, [高]{●}等学[校]{●} is abbreviated a bit oddly, as [高]{●}[校]{●}
Reason for using 高校 as the abbreviation for 高等学校
I cannot find anything definitively explaining why 高等学校 was abbreviated as 高校 instead of 高学, but @naruto's guess seems like a good one -- there are many homophones for こうがく, several of which appear to be in common use even in the spoken language, and most of which already end in 学 in the written language. While there are also many homophones for こうこう, fewer of these appear to be in common use in spoken Japanese, and the only one with the 校 character is 高校, which may leave less room for ambiguity.