Although the term "transfer" is commonly used, in actual fact in all so-called visa transfers what is actually happening is that the original visa is cancelled and a new one for the remaining time available is issued for the new employers. H1 visas are employer specific and cannot technically be transferred at all. The term, transfer, is inaccurate and is used as a convenience, but it does not reflect what is actually happening.

There would have to be a new visa issued under the new employer, which would be for three years minus whatever time you have already used. There is a one-time three year extension available but if you have already used it on the first visa you would not get another one on the new visa - the six-year limit is to the employee, not the visa.