Words or phrases may vary based on how they are use in a sentence. Let's say for example, the word "see".
I see his dog barking in front of your store last night.
I see him as a brother not as a boyfriend.
In my first sentence, the word see literally means something that was perceived or spotted by the eyes, while on the second part of the sentence the word see was used as something you have discerned or regarded as.
So I think the same goes for this situation, it is based on how the phrase "take a second" was used in the sentence to conclude whether or not this phrase should be accompanied by the word "look". However, judging on how "take a second" was used, the author was just trying to suggest his readers to take a pause or break and think things over before speaking or writing to avoid committing errors and to make your statement clearer.
Besides, when you say "take a second look" it basically means the phrase itself: taking a look on something for the second time or reexamining things or having things double-checked.