I don't think he needs more good advice. He's had loads of that from post #2.
He needs to get off his bottle & glass and follow some of it!

Your output from the parted -l command shows an msdos partitioned drive which generally would use Lilo. It is possible to do an EFI install on an msdos drive but not the recommended way.

Mount /dev/sda1 and take a look at the contents. It should have a Slackware directory if it is an EFI install. If it isn't there, you don't have an EFI install.

If you had an EFI install, it should show an entry in /etc/fstab for /boot/efi which you do not have.

The kernels are in the /boot directory using a Legacy install and on the EFI partition with EFI.

If you have problems after updates of the kernel, try updating grub (as root) after the update and before rebooting with:

Does the contents of boot partition look as it should: yes
Boot mode: legacy/csm
drive type gpt or msdos: msdos
partition layout, filesystems, mount points, the existence of an efi partition: no.


What is odd: The first partition looks like it may have been a windows efi partition at one time that was converted to swap.
As per Yancek, did you run grub-mkconfig after updating. Slackware doesn't auto update grub.cfg when kernels are updated.