You need to apply for a transfer of sponsorship. The process is essentially the same as a new application - you need your new sponsor to justify why they need you and show that they can't get a Permanent Resident. The only thing you don't need is to resubmit all the copies of your qualifications and so on. The form is the same one:
http://www.immd.gov.hk/pdforms/id(e)936.pdf
You may not work for the new employer until you have the approval from ImmD. In my case when I did this about 3 years ago it took 3 weeks for the application to be approved. If your visa has only a few weeks or months to run (as yours does) then they will issue a new visa, but if it has more than 6 months to run then you will just get an authorisation letter.
In fact in your case it seems unlikely that you will get the transfer approved before your visa expires, so you will also need to do something about that. Options are:
1) apply for an extension of stay on the grounds that you have applied for the transfer of sponsorship
2) exit and re-enter as a visitor until the application is approved
Don't ignore this - Immigration are quite strict on people who over-stay their visas, and it would harm your chances of getting the application approved. I suggest you talk to the Immigration people about this when you submit the application and ask for their advice.