There is NO "next smallest integer that is a perfect cube less than -1".


It sounds strange, but let's look together.


There is -8 = .  It is an integer, a perfect cube and less than -1.


There is -27 = .  It is an integer, a perfect cube and less than -1 and less than -8.


There is -64 = .  It is an integer, a perfect cube and less than -1 and less than -27.


There is -125 = .  It is an integer, a perfect cube and less than -1 and less than -64.


       And so on . . . . . . . .


For any such example and for any such given integer, there is another integer, which is a perfect cube 
and less than -1 and less than the given integer.


It exactly means that THERE IS NO the smallest such an integer.

I looked in GOOGLE, and Google explained me that


    - the term "next largest" means "the second after the largest", i.e., in an ordered list in the descending order, "the second from the top".


    - "next smallest" means "the next greater than the smallest", i.e., in an ordered list in the descending order, "the second from the bottom".


Now, if you apply this knowledge to your problem, you will see that in this problem, there is NO such a number, 
which is "next smallest integer that is a perfect cube less than -1".


There is no such a number, simply because there is no "bottom" in this (ordered) list.


There is  "next largest" in this problem, and it is -8 = (-2)^3,
but there is no "next smallest" in this case.


This prose was in use, probably, at the Shakespearean time and at the era of steam and iron.


In modern life, nobody uses this language.