I figured it out. I was just thinking backwards in my solution.
Instead of my initial case function, I was able to use the following:
case
when [Social Security Number] between 0 and 9 then '00000000' || cast([Social Security Number], varchar(10))
...
Instead of my initial case function, I was able to use the following:
case
when [Social Security Number] between 0 and 9 then '00000000' || cast([Social Security Number], varchar(10))
...