What I did was install Mint on /dev/sda6, and tell it to boot from sda6. It's thick enough to take that from you
It won't boot, of course, until you copy over the kernel, system.map & config to your slackware /boot and add an entry to grub or lilo. Then all is dandy, and when it updates stuff, it's harmless. Just remember to update the real /boot occasionally.