Tier-2 visa: Start date problem

Hi !!

I graduated in December last year and my visa was to expire by the end of January. However I applied for visa extension before that. Now, I have got a decision of refusal against my application and have been asked to either leave the country or submit a fresh application within 28 days. I have now found a job in a Graduate role in an A-rated company that is willing to sponsor my Tier-2 visa. The starting date of the job is in the end of September. I have just signed the contract but don't have the CoS yet.

The problem is that I can't apply for the Tier-2 visa before the end of June (3 month rule) but I wouldn't be able to stay in the country for so long. Could anyone tell me a solution for this please? Has anyone faced a similar problem before? Is it possible for me to apply for Tier-2 from my country. I know that it might be a little complicated because of the Resident Labour Market Test (RLMT). Is it possible for the company to give an early start date on the CoS? This is a really difficult period for me and any help on the issue would be much appreciated.

TIA