How to remount external drive in OS X El Capitan without rebooting?

You can, in Terminal, try:

sudo diskutil list

Check the output for any drive matching the external drive you have connected. /dev/disk0 is your internal drive. If you have no other drives connected, then your external disk should show up as /dev/disk1.

If you can see the drive, run, from Terminal (assuming your drive is /dev/disk1):

diskutil mountDisk /dev/disk1

This should properly mount your external drive.

Open the terminal.

If the disk is not encrypted

Enter this into the terminal:

diskutil list

It will list all the disks that are physically connected to the computer, even if they aren't mounted.

Look for the name in the list and the corresponding code in the format disk#s#. I'll pick the code disk4s2 out of the air, you use what is in your list. Then mount the disk with:

diskutil mount /dev/disk4s2

If the disk is encrypted

I encrypt all my disks. If the disk is encrypted, you must use CoreStorage. It's just as easy, except you have to slog through my long-winded explanation, and you must use the disk's Jumble Number instead of its disk#s# designation. I call it the Jumble Number because it is not labeled and it is a long jumble of numbers. Don't faint. You can just highlight it, copy it with CMD+C and paste it with CMD+V.

First, find the disk:

diskutil coreStorage list

The word "coreStorage" is case sensitive. You must type it in lowercase but capitalize the S, otherwise it will say "command not found."

Look for the name in the list (for example, "Fred's Disk"), but this time use the Jumble Number that is directly beneath it.

Mount the disk as follows, but don't use the Jumble Number in my example, use the one you find in the list:

diskutil coreStorage unlockVolume 29B8Ad4C-2608-4822-A272-3DA848CACB83

Someone who is handy with AppleScript can probably whip up a substitute for Disk Utility that actually works.