The problem that you are experiencing seems to be a problem recognizing the hard drive and this is why it takes a long time.
The e-SATA hard drive is self powered?
Is it tuned on a few seconds before you turn on the computer?
Did you tried to update your motherboard BIOS?
Have you tried to set a hard drive delay in BIOS? (this option is not on all of the motherboards, you will need to check with your system manufacturer if yours has it)