Why not generalise this a bit, to a subroutine that can adjust a given time by a given number of seconds? And rather than using an if to determine whether you need to pad the hours with a leading 0, use sprintf to format the integer to a 2-digit string like so:

#!/usr/bin/perl
$now = '00:00:00';
for (0..60*60*24+5) {
    print adjust_time($now, $_), "\n";
}
exit;

sub adjust_time {
    my $start_time = shift;
    my $increment = shift;

    my ($hour, $min, $sec) = split /:/, $start_time;
    my $seconds = 3600 * $hour + 60 * $min + $sec;
    $seconds += $increment;

    $hour = sprintf "%02d", ($seconds/3600) % 12; # add + 1 here for 1
+2 hour format;
    $min = sprintf "%02d", ($seconds/60) % 60;
    $sec = sprintf "%02d", $seconds % 60;
    return "$hour:$min:$sec";
}
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