Hi Derek,
it's me again. Instead of spreading words of wisdom, let me actually try to help you, if your problem still does exist:
After you have clearly installed your system und turned on the external TM-volume, you will be asked whether you'd like to use this volume for back ups or not. Do select "NO" (or anything equivalent – I am sitting in front of a German user interface ;-), and keep the external volume mounted.
Open your System Preferences, go to Sharing (third row, last item), left-click it and change the device name into the one you want, but choose a name without special characters, punctuation, and blanks (e.g. DEREKSMAC). Confirm the change and close the System Preferences. Press CMD-ALT-SHIFT-Q to instantly log-out, then log-in again and start with the following procedure:
Now, press CMD-SHIFT-U in order to open the Utilities folder, then double-click "Terminal" so that a window with a DOS-like prompt appears. In the following, I will make use of the ampersand-sign (&) in order to mark those parts of the commands where you will have to replace the "dummies" with the names of your very own volumes, folders and so on. DO NOT USE THIS SIGN WHILE EXECUTING THE COMMAND IN THE CONSOLE WINDOW, OK?
In the active(!) Console window you will see something like ~ &YOURUSERNAME$ followed by a grey rectangle – that' s the input-prompt.
Now, type ls -a and press Return –> the content of your user folder will appear; some file start with a dot which marks them as invisible. Just ignore them!
OK, now type cd /Volumes/&NAMEOFYOURTMVOLUME followed by return -> you will see &NAMEOFYOURTMVOLUME &YOURUSERNAME$ which means that you have changed the directory to the first level of your TM-volume.
Type ls -a again, and voilá: there's a folder called Backups.backupdb
Type cd Backups.backupdb, press Return, then type ls -a, press Return again -> you'll see only one folder without a "starting-dot" which has the same name your Computer was given above, e.g. DEREKSMAC (you remember?)
Type cd &COMPUTERNAME, press Return, type ls -a, press Return again -> You'll see on folder called "Latest" which is an Alias / Link to the last incremental backup which successfully stored by Time Machine. The other folder names have the format YYYY-MM-DD-HHMMSS. Look for the one with the date which is closest to the lucky times before(!) you installed El Capitan for the first time, this is your &DESIREDFOLDER!
And now, let the magic beginn and pray :-)
ATTENTION: From now on, I won't be responsible for anything going wrong; do it on your own risk if you trust me or ask somebody else to assist you !!! Double-check the input-phrase for misspellings in order to reduce the risk of errors !!! Do not forget to replace the &VARIABLES with the actual names of your files and folders !!! You can cancel here by typing cd , then press Return and start again when you want to write down the names of volumes and folders, OK? Or just think like Pliny the elder while heading for Mt Vesuvius:
Fortune favours the brave ;-)
Type sudo ditto -rsrcFork /Volumes/&NAMEOFYOURTMVOLUME/Backups.backupdb/&DESIREDFOLDER /Users/Shared and press Return. You will have to type your (admin) password without seeing anything: neither what you type nor how many sign you already have typed. Press Return again and wait until the ditto command has copied the content of your selected backup-folder into the Shared folder on your clearly installed internal volume. The successfully ending of this procedure is marked by the input-prompt reappearing.
Now you can check it by typing cd /Users/Shared followed by Return, then type ls -a and press Return again -> You should see the whole content of your former Home folder with all your data in the Shared folder!
Type cd and press Return, type exit and press Return. Close Terminal by CMD-Q and unmount your TM-volume.
Open your new Home folder by typing CMD-SHIFT-H, press CMD-(ARROW-UP), select the Shared folder and open it by double-clicking. Open a second window or Tab which displays your actual Home folder and manually move or copy the desired items (the CONTENT of the Documents, Pictures, Music folders, not the folders themselves!) in to their equivalents in the second Tab or window.
I'd be happy to hear from your being successful.
Kind regards and do excuse my "buggy" English!
Stephan