Hi JBG,
Sorry I have not replied sooner, been busy.
Are you still trying to get the hard drive install to work?
I guess you want to keep Linux mint?
Tiny Core Linux is two files. core.gz which is the file system and vmlinuz which is the kernel.
Both are needed and both are on the CD.
You would need to boot into linux mint then run the CD to copy the files to a directory(folder) created on the hard drive.
ideally in / , the root directory, but you would need to sudo in mint to do that. perhaps name it tiny_core_linux.
To add tiny core to the boot menu it would be necessary to edit the /etc/grub.d/40_custom file as root.
This may take some explaining but I believe the code that Leee provided would work then run the command to update grub.
Hope this helps some,
Joe