Rainboe, it is the TEMP nav database that gets automatically deleted after flight – hence the name; the SUPP database will remain.
To delete the SUPP database: From INIT/REF type SUPP in the scratchpad and then line select NAV DATA>. You should now be on the SUPP NAV DATA page, line 6R has a prompt DELETE ALL SUPP DATA> which will do the trick. Note that this is only available before you have entered an origin airport.
S&L