Changing employer after current tier2 visa ends

Hi,
I'm on an open contract and has just been informed that by the end of May it will not be renewed. My tier 2 visa with this employer ends half a month later. My questions are-
1. For how long am I allowed to stay in the UK after the visa ends? Or do I have to leave by the end of the visa date?
3. Since this is an "open end" contract, in case my employer can support me for a couple of months more, do I need to apply for a visa for such a short period? Or is there another possibility since this is an "open end"?
2. Given the "cool off period", If I find a new employer in the UK, does the new contract have to start while my current visa is still valid, or else I'm not allowed to work for a year in the UK? Or is it enough the new employer provides a CoS before the current contract ends, even if the work itself starts, for example, only in August?
Many thanks for your help

1. You must leave the UK by the latest, the date your visa expires. Once you are no longer employed, your employer has an obligation to notify HO and your visa will be cancelled.

2. You have to apply for a Tier 2 visa extension to be able to work. You will have to meet all the same requirments again. You can only work for the new employer once a new visa has been granted.

3. You need to find an employer and apply for a new visa before your current visa expires. If you cannot find one, you will have to leave and yes, you will be subject to the 12 month cooling off period.