Basically, the procedure is as follows:
Install your new distro, eg Slackware, without (skip) installing a bootloader
The following steps are done in your original distro (the one with the working bootloader):
Make a directory in /mnt called "Slackroot"
Edit /etc/fstab to load your new Slack root partition under /mnt. Example:
/dev/hdb1 /mnt/slackroot ext3 defaults 0 0
Naturally, you would change the information to suit your setup
Edit /etc/lilo.conf and add an entry for Slack. With the above example, it would look like this:
image = /mnt/slackroot/boot/vmlinuz
root = /dev/hdb1
label = Slackware
Save and run (as root) "lilo -v"
That's it. Please note, this is how I do it, I'm sure there are other methods. Also remember some distro's will need different entries in lilo.conf, depending on the kernel image and where it's stored.