You need to find out what boot loader is installed on your boot partition.
If you don't get any useful output from running the grub command, it doesn't mean that grub isn't installed on your boot partition but only that you don't have the files to install grub currently installed on your running OS.
Actually, Rich's advice to check for extlinux.conf makes the most sense, as syslinux is the default boot loader installed by Core when using tc-install, IIRC.
syslinux uses extlinux.conf by default if installed on ext2/3/4 filesystems but it also searches for syslinux.conf if the former wasn't found.
Also, boot loaders search in more than one directory for their config files.
Common places are, depending on the boot loader...
also depending on the boot loader...
Try to find one of these or similar and check if a Tiny Core related entry is contained.