Perhaps you could simply keep track of the file position of the last log entry you've processed, and then process all entries found after that. If the file position is smaller than the previous day, then take all the lines. Something like (untested top-of-the-head) this:

# Read logfile names with last position read yesterday
open $inf, "<", "loglist.txt" or die;
while (<$inf>) {
   chomp;
   my ($fname, $fpos) = split /\|/;
   $logs{$fname} = $fpos;
}
close $inf;

# Get new lines from each file
for my $fname (keys %logs) {
   open $ouf, '>>', $fname . ".cumulative" or die;
   open $inf, '<', $fname or die;
   if ($logs{$fname} < stat($inf)[7]) {
      # Continue from where we left off yesterday
      seek $inf, $logs{$fname}, SEEK_SET;
   }
   else {
      # start at beginning of file
   }
   while (<$inf>) {
      print $ouf $_;
   }
   $logs{$fname} = tell $inf;
   close $inf;
   close $ouf;
}

# Rewrite list of files and positions
open $ouf, ">", "loglist.txt" or die;
print $ouf join "\n", map { $_ . '|' . $logs{$_} } keys %logs;
close $ouf;
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