Z-visa to stay permit to then extend L-type

Hi there
Can anyone help?
I am currently on a z-type and working, but plan to terminate my contract Mid-April. My school will change my z-type to a stay permit (is that an L-type???) for less than a week (which would mean I need to leave the country).
What I want to do is stay in the country a little while longer, as I have another job which will start in August. However, I believe at that point I would need to leave the country to obtain a z-type.
Is it possible to extend the stay/L-type for 30 days? I live in Hangzhou, Zhejiang Province. Can I do this at the local PSB? or do I need to go to Hong Kong to do this?
Thanks