my %slot_lkup =
    reverse
    map /^Slot address (\d+).*\n +Volume Tag[ .]+([^\n]+)/s,
    split /(?<=\n)(?! )/,
    $inventory;

...
$slot_lkup{$vol}
...

##</code><code>##

my %slot_lkup;
{
    my $slot;
    while (<$fh>) {
        if (/^Slot address (\d+)/) {
            $slot = $1;
        }
        elsif (/^ *Volume Tag[ .]+([^\n]+)/) {
            $slot_lkup{$1} = $slot;
        }
    }
}

...
$slot_lkup{$vol}
...