Get the slackware packages from one of the mirrored sites listed from slackware.com or go to www.linuxpackages.net
to get the latest ones.
Then instead of removing the packages and then reinstalling, you can do something like this:
Download the packages and place them in /var/log/packages
This is where Slackware keeps the listing of all the slack packages installed.
Say you got apache installed, version 1.3.27 that came with your version, you would do something like this to upgrade:
# upgradepkg apache-1.3.27-i386-1%apache-1.3.29-i486-2.tgz
Or something along those lines. Its basically in this format:
# command [old-installed-package-name]%[new-package-name.tgz]
If that's too daunting, just a simple:
But this only works with slackware tgz files.
You can also look into using www.swaret.org