No, and a qualified yes.
There is no such thing as transfer of an H-1B, although some do refer to it as such. Nothing is transferred, not by one employer to another, and one doesn’t even have to know about the other. What moves from one to the other is the individual with the H1-B authorization.
In actuality, a new H1-B petition has to be filed the new employer, and it may be without the restriction of the cap. And, as with your previous filing, you would need to re-submit your documentation, as appropriate.
As you are already in the US on a non-immigrant visa as an employee of an international company with offices in both the US and abroad, you may want to seek the guidance of an immigration attorney in advance of a new petition.