Hey, time to use my favourite trick: opening a pipe to a post-processor:

#!/bin/perl -w
use strict;

my $log= "logfile";

open( STDOUT, "| perl -p -e 'print q{[}, scalar localtime() , q{] };' 
+>> $log" )
  or die "cannot redirect output (open): $!";

print "Here he comes to save the day!\n";

# warning, if you don't have permission to write the file
# then the open will succeed but the close will fail, 
# try it, it's quite surprising
close STDOUT or die "cannot redirect output (close): $!";
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