The problem is the term “transfer” which is generally used on several of the forums when someone is in H1B status, files a petition for a new employer and is able to begin work based on the receipt of the new petition at USCIS. You can basically work before receiving approval of the petition. The paystubs from the old employer are an easy way to demonstrate that the employee is in status because they have been working for the old employer.
If someone is not in H1B status at the time of filing the new petition, he must wait for the approval notice before beginning work on a change of employer. A few people also refer to this as "transfer".
Both involve new filings from your employer. It is essential that you understand the acceptable start date in the two situations - based on the receipt of the petition verses approval of the petition. Someone with a first time approved petition this year cannot be in status before Oct 1, thus the statement that you cannot "transfer" before Oct 1 is true when used in the first context.
Obviously, you can change your employer on OPT at any time since your OPT is not limited to a specific employer. An H1B is tied to a specific employer and so the problem will arise on Oct 1 if you do not have an H1B approval with I-94 for your new employer to supercede the H1B I-94 for your old employer. The safest thing is to ask your new employer to file the H1B petition in PP so you definitely will have the new approval before the Oct 1 start date.
If you have not received the new notice, you will be expected to be employed by your old employer beginning Oct. 1. If it gets to be September 10th or so and you do not have your new I-94, your employer's lawyer should be contacted for further advice.