You're correct when you say that your EP is based upon your 'new' company and you as an employee - you have to be sponsored by an employer to be issued with an EP.
I came to Singapore with one opportunity only to see it disolve inside 3 months - result: return to the UK or find another opportunity here.
I found an alternative and have just been through this process. Your existing EP is cancelled upon leaving your existing exployee (as he/she can no longer sponsor you) and you must re-apply in full, with all of the documents, 2 x photos, etc., etc., you originally required. There is no guarantee that you will be granted an EP with a new employer, though I have been told that existing EP-status can help - hence, try to co-ordinate the cancellation/re-submission process into one movement.
The EP stamp in your passport will be cancelled and the PP returned to you pending re-submission. Thereafter, you have xx days (usually 30, I think) in Singapore as a visitor but with the right to work.
I believe that an employers' status vis-a-vis industry segment, established nature (or otherwise), size, etc., all have a bearing upon how likely you are to be re-issued with an EP - and your own skills will also be important. 'In demand' experience is often appreciated, apparently. Your 'new' employer should be prepared to do much of the donkey-work in chasing the MOM and settling any queries that may arise - especially if you're being head-hunted!
Whilst it's not a formality, I'm not sure that you should be over-worried by the prospect of moving jobs. If the same skills which brought you to Singapore and saw you being granted an EP in the first place are presumably still to be brought to bear in a new position with a new employer, I suspect that you won't have too much to be concerned over.