Hi JBG,
Sorry for the delay.
I found a DVD from a magazine with Linux Mint 18.3 and installed it to my older laptop. Also burned the core iso to a cd. This way I can follow what you are doing and hopefully be more useful in posting.
First off is the Mint issue.
Do you intend to keep and use Mint, keep it updated?
Do you know the password, can you access the root account?
In order to install Tiny Core along side Mint it would be necessary to boot from Mint. This would require editing Grub2 in Mint as root.
It is possible to boot off the CD and have some persistence, retaining your changes, this is very basic and others here may have a better way of doing this.
Boot the CD
Mount the hard drive, which I am guessing is sda1, the first partition of the first disk.
sudo mount /dev/sda1 /mnt/sda1
create a directory called "tce".
sudo mkdir /mnt/sda1/tce
reboot
check to make sure it worked.
ls /mnt/sda1/tce
There should now be items in the directory and the hard drive was mounted automatically during boot.
Changes made in the home directory, /home/tc and the opt directory, /opt can be saved before shut down.
When a program is downloaded using the app browser, tce-ab, choose on-demand so it can be saved to the tce directory.
The system will not issue a prompt to back the system as it does from the GUI. So it is essential to remember to do that manually.
To backup the system before shutting down.
filetool -b
I believe this will automatically restore saved data but would have to do more testing to confirm this.
I can walk you through installing to the hard drive and editing Mint to dual boot, as I did that today on my laptop.
If you intend to stay with the command line interface, it would be helpful to become familiar with the basic shell
commands like ls , cp , cd , etc. There are many resources on the internet for that.
Hope this helps,
Joe