listen carefully, you haven't told the date your ARC expires, which would help, but be very wary of trusting your old boss and new to get this done in the correct time.
The basic process, even though I am unsure of the dates...
1. Get your health check done asap
2. Get your photos, photocopied passport and degree ready asap
3. Get your contract from your new school ready, to start from the same day your old ARC expires, for 1 year, minimum salary say 50k
***You might also get a cancel contract form from your old school on your last day of work, that's a side issue. See tealit.com
4. Get the new school to prepare necessary documents in Chinese (contract, others-I'm unsure exactly what they need)
5. Take them yourself a week before your old ARC expires (or make sure your new school does this. MAKE SURE OF IT, don't just trust them) to the Labour Bureau
6. Labour Bureau will (hopefully) accept the documents with 500$ fee, and give you a receipt.
7. Take this receipt to the Immigration police and ask for 1 month extension of your ARC to process the new ARC.
8. After 3-4 days, pick up the approved application from Labour Bureau and take it to Immigration Police.
9. Apply for new ARC
10. After 2 weeks, receive new ARC.
good luck.
A pessimist sees the difficulty in every opportunity; an optimist sees the opportunity in every difficulty.
- Sir Winston Churchill