Exiting USA on a different passport from the one I used to last Enter

Hello,

I am currently in USA on L1 status. Is it possible for me to depart the USA using a passport or identity card different from the travel document I last used to enter the USA?

I entered the USA using a foreign refugee travel document with a valid US visa. My I94 and my ‘admitted’ stamps are on that travel document.

Now I have a new passport ( it is not that same type refugee travel document).

When exiting USA, do I need to show or use the refugee travel document I used to enter USA? What problems may happen if I am not able to use the old travel document anymore?

I do not have it and cannot use it at this time and I am wondering if I can travel outside for stamping the new US visa all using the new passport.