As long as your H-1B is current, you can changer employment as soon as the petition is filed and the USCIS issues a receipt to the new employer. So the earliest you can begin working for your new employer is the date on which the USCIS receives your H-1B transfer petition.
Your employer will need to file an LCA (Labor Condition Application), which can take from 1 to 2 weeks. After the LCA is approved, your employer will send the petition package to USCIS where it will take anywhere – on average – from two to six months to process (sometimes less, sometimes more depending on how busy they are and whether they need additional documentation from you).
If you pay an additional fee of $1,225 for premium processing, your application will be substantially expedited – usually processed within two weeks, give or take.
Thisto the USCIS site allows you to check your status using your receipt number.
For help with answering any additional H-1B questions or processing an H-1B application, check outfor cost-effective help from top-notch attorneys who are experts in immigration law. Also feel free to PM me directly.