My husband has a big problem with lying and substance abuse. I think he may have a second cell phone to help him with his dirty work. Will a background check show me phone numbers associated with him? If not, is there another way that I could perhaps search phone records with his social security number?

No.. ever hear of prepaid phones?. and you would need a court order for some of those records anyway.
For the very same reason as if a stalker down the street wanted to do checks on you without your knowledge.
Not everything is linked to a SSN and that's a good thing.

I agree with smoothy. A cell phone that is kept charged somewhere else, like work, is easy to hide. In addition, there are forwarding services that can allow giving out a number that has no appearant relation to your phone, but calls can be forwarded. One phone - multiple numbers - even harder to find out about.