Edit: I didn't realize you are pretty experienced and probably know most of this already.
TL;DR: root device, install Stickmount via ADB, profit. Physically speaking, grab a USB-OTG cable and use that to connect your device. Probably a bad idea to run any hard drive that isn't externally powered, but flash drives work great.
So I don't have one of the devices physically, so I can't guarantee that this will work. However, I have fiddled with one of my friend's devices separately, and have done extensive work on my FireTV, so I'm fairly optimistic this will work.
Your first step will be to root the device. This means that you have access to all parts of the OS that Amazon was previously restricting. This will allow you to install Stickmount, which is an app that enables your device to mount external storage.
Once you have downloaded the APK, follow the XDA-Developers page on installing Minimal ADB and Fastboot. This will be used to connect to the android device.
You have a couple options on how to connect. You can either use a USB cable, and there's lots of documentation for that, or you can connect via wifi. I find wifi connection to work fine, so I'll leave instructions for that method.
1) Drop the apk of stickmount into the same folder as adb.
2) Run adb per the instructions for your machine from XDA-Developers.
3) Find the IP address of your Kindle.
4) Run the following commands individually:
adb connect (IP address of your device- probably 192.168.1.xxx or thereabouts)
adb install (filename of Stickmount- probably something like stickmount_xxx.apk)
If successful, Stickmount will have been installed on the Kindle.
Run Stickmount to make sure it works, then restart your device. On reboot, you can plug in the external storage via a USB-OTG cable. I'd strongly advise against using a self-powered hard drive- I'm not sure how much power SSD's draw, but a spinning drive is definitely out. You're totally safe with a flash drive or card reader though. I did this a lot with my Nexus 10 on a road trip- worked great.
Once rooted, you have a couple other options. I won't clutter this response here as they are unrelated, but you can install the google play store, use XBMC/Kodi, and a lot more. Feel free to ask any questions you have about that.