I'm cuban Working in China

Jessica, you can just apply for a tourist L type for an additional stay in PRC. You can apply for the L type 7 days earlier before the work Z type expires and it will take 7 working days. It will cost CNY 168. You can just try to do that at the exit and entry administration of the local PSB within PRC. If it cannot be done, you can go to HK or Malaysia for a try.

If you apply for a visa after your residence permit for work, it is exit visa, not exactly L visa.
With the exit visa, you have to leave before it expires. You cannot have it extended.
What's more, exit visa is quite short usually. 10 days, 14 days, or the longest 30 days.
You can ask the company to extend your residence permits for some months.