To change employers you need to keep your passport's Z visa valid and get all the paperwork necessary with you. One of those documents is the release letter and the other may be the recommendation letter. On top of that, you may need the foreign expert evaluation paperwork (one form filled and stamped). Note that I have said "may" to two docs for the reason that not everyone does it the same way in the country.

As for your suggestion to stay on L visa, you should not convert to this choice because it is why you may have to leave the mainland to re-apply for the Z later. One of my colleagues had to travel to HK to get his Z prior to his Residence for work that he got later then.

Anyhow, 10 month contracts have their pros and cons; if your legal work permit expires before September when you are to start the new gig, you should ask your new employer to cooperate in order to keep you in the country. If your new employer does not cooperate, find another one who does. To avoid such nuisance in future, be firm on your request that you need your residence permit for 12 months to stay.