I guess lot of us are in the same boat and everyone will stumble upon this question.
As far as the limited knowledge I have, I don't think it is possible, because as the general rule goes " A GC is restricted to per position per person/family per location per company" what this means is: A GC in an employment category is issued to a person and his dependants for a particular position located in a particular geographical location for a particular company.
On the other hand a person qualifies for a H1 7th yr extension only if he/she had applied for a GC labor on or before 365 days of the current H1 being subject to expiry.
Hence if you are willing to transfer from one company to other, then your current GC will become invalid and you do not qualify for an extension.
That said, there are several loopholes through which one can get away.
If anyone got an H1 transferred successfully after the regular 6yrs quota is completed please respond.
Thanks,
-DR.