Did you do the original install, or someone else? Or I wonder if there is another lilo.conf in a different place or with a different name. But doesn't really matter now!
A few things to make yourself a note of, for future reference (while fresh in mind):
1. You can always boot into the install disc to troubleshoot your Slackware install - no real need for another boot disc.
2. Once booted to the installer, you can mount and chroot into your actual system to do things like configs - and reinstall lilo, or to recover data from a failing drive.
3. It would be easy to start by just reinstalling lilo, but a systematic check of drives and boot configs is always best - now you know why it didn't work and also know your drive is OK. Best confidence route!
As for grub - I am not a grub fan or user. As long as you are on a Slackware install with a machine that supports it, Lilo is MUCH easier to manage and troubleshoot, in my opinion! Just stick with what works!
I have a few additional comments I may try to add later but out of time for now!
Glad it works and good luck!