It is not your computer's OS that defines the ESSID... it is the Access Point. Whatever the Access Point has as the ESSID, you need to configure in your rc.inet1.conf
To give an example - suppose that in rc.inet1.conf, the definitions for your wireless interface have the index
[4], then you add below something like this:
Code:
WLAN_MODE[4]=Managed
WLAN_ESSID[4]="YOUR AP's ESSID"
WLAN_WPA[4]="wpa_supplicant"
WLAN_WPADRIVER[4]="wext"
Those lines will also trigger a start of wpa_supplicant and provided you correctly configured your /etc/wpa_supplicant.conf you will run a secure wireless connection in no time (use the example "wpa_supplicant.conf" file that Slackware installs and enter your own PSK/ESSID information).
Eric