Once you download the new kernel source code .xz package.
mv/cp to /usr/src/ and unzip/package it there(/usr/src), you will have your new kernel source directory.
and then remove the symbolic link of older (linux-version-number), re-links to the new kernel (linux-newer-version#) to linux.
rebuild your new kernel in your new kernel directory not the old, computer is smart, but it can only do what you tell them to do, it can't guess what you might likely to do.
inside of new kenrel directory: make distclean
get official slackware kernel configuration file ( example to build generic-3.10.17 for slackware64 )
copy this file into new kernel(3.10.17)directory as .config file
make ; make modules_install
inside it contains 3 directories, using inside kernel-generic slackbuild scripts to build a slackpkag for installation.
don't forget using kernel-modules's slackbuild script to build kernel-modules of same version.
you need huge package too be on safe side, it for trouble shooting.
DO NOT use upgradepkg to upgrade kernel. Use installpkg
edit your /etc/lilo.conf add your new kernel. keep your old kernel intact, just be safe, testing your new kernel first.
go to /boot; check and make sure links are good.
mkinitrd with your new kernel, add necessary modules into your initrd ram disk.
13.37 version of lilo and mkinitrd don't work with 3.10.17, you need current version source and reompile them with glibc libraries of 13.37
If you using LVM, you might need new version of lvm2 and udev as well
run lilo before your reboot.