Second/side job while on L1

I'm back on a track!
Finally I've got my L1A and now I'm thinking about having a second/side/contract/evening job.
I've read related topics about second job with TN- does anybody know the difference - what kind of primary visa (TN,H1B,L1) do you have.
It wasn't very easy to obtain L1 and I'm gonna apply for GC soon so just want to know is it legal at all and what the downsides of my activities outside L1.

Thank you Steven for your advices and Reba for your posts in that harsh time of my relocation to US this summer. (L1B deny from USCIS-TN- then L1A at POE)

You can't hold two categories at the same time. You can have multiple TNs (which in reality is one TN-1 with multiple employers) but not L-1A and other categories I'm afraid.