Searching through a whole lot of PDFs on an external disk
Part of it might be a setting in your energy saver settings that allows hard drives to sleep if not in use, you could check on that setting and see if adjusting it makes the job easier and searches quicker.

A USB 3 drive (assuming it is on a USB 3 port) should be plenty fast enough to handle the necessary bandwidth. What might be slowing you down is the speed of the drive. If you have a 5400RPM drive that would slow things down and moving to a 7200 might make it fast enough for you. But fast enough is hard to quantify.

Thunderbolt on SSD would be as fast as an internal SSD, but these drives are pricey and when I was looking for one recently found none that were suitable. Many being multiple drive RAID chassis. Instead I went with a Firewire 800 drive case (from OWC) and a Samsung 1TB SSD.

As my Mac did not have USB 3, Firewire 800 was fast enough for my needs. And if you already have a USB 3 case putting an SSD in it might be your best bet for maximum speed and affordability...

Unless of course you spring for an external Thunderbolt RAID chassis...