Does this pronunciation follow a rule of some sort? I couldn't find any answers online about this example specifically.
There are several websites saying that reading 木 as こ is an example of 転音.
- 木 alone is always read as き
- 木 + 陰 is read as こかげ
This chiebukuro answer lists such words. It seems much more common for surnames (e.g. 木暮=こぐれ). (NB the linked answer doesn't say it's a 転音.)
Personally I have some doubts き→こ is really an instance of 転音. First, I guess a large majority of native speakers would read 木の下 as きのした. (Not that このした is wrong, there is a dictionary entry.)
A few other words with 木の..:
Of these, 1 would be read mainly as このみ, while きのみ is perfectly acceptable. As for 2, I think きのめ is more common than このめ. Anyway, these two 木 have both readings of き and こ, as 木の下.
A written form appearing earlier may not prove everything, but it should be safe to say that こ reading is just as old as き.
Based on these, my opinion is that both き and こ have existed as very old native Japanese words, and somehow combined with the introduction of the Chinese character 木. Hence the answer to the question is 'that is just the way it is'.
When the two words overlap, the pronunciation of the second words change. This rule is called 転音 (ten'on).
Umbrella: 雨 (ame) & 傘 (kasa) = amagasa, not amegasa
Sake barrel: 酒 (sake) & 樽 (taru) = sakadaru, not sakedaru