If you are in the U.S. with an approved H-1B and if you have an I-797 Approval Notice with an I-94, and have a job offer from another employer who will be filing a "change of employer" H-1B petition on your behalf, you do not need a visa stamp in your passport. Please consult an immigration attorney to discuss this further,
Disclaimer: The information provided here is generalized and should not be relied upon as legal advice. This communication does not create an attorney-client relationship.
The new company will have to apply for an H-1B first. After it is approved you may transfer. Regarding whether this can be done before October is a question to address with one of the immigration lawyers handling the two H-1B petitions.
The answer provided here is general in nature and does not take into account other factors that may need to be reviewed for a more precise answer. You should consult with an immigration attorney before taking any action. The answer here is not intended to create an attorney-client relationship.