Unfortunately, there is no way to expedite the port request. Contacting your current carrier before submitting a port request is the best way to ensure a streamlined process. You'll want to confirm which billing name and address need to be entered on the Letter of Authorization (LOA). If the number is wireless, please also verify the account number and pin with your carrier.
Do not schedule a date to cancel service with the carrier until after the number has ported successfully. Premature cancellation of your number may cause the porting process to abort.