Coming from the word 少（すこ）し
(a little bit), すこしも～ない
is an expression which translates almost exactly to 'not even a little bit (A)' in English. Due to this, it indicates that there is 'no (A)', '(A) never happens', or 'not (A) at all', depending on whether a verb or an adjective is the word that is being negated. すこしも
can come at the beginning of the sentence, or directly before the word/statement it is referring to.
There are a lot of people today, so don't stray off. Not even a little bit.
What is this cake, it's not even a little good!
Her boyfriend is not even a little handsome.
This construction is another example of how the も
particle expresses that the word before it is surprising in some way, or unexpected. This was briefly discussed in the Number + も
Here we can see that も
is highlighting 少（すこ）し as being 'surprising'. This is where the 'not even!' meaning comes from.