use File::Tee:

use File::Tee qw(tee);
open my $LOGFILE, '>>', $logFileName or die $!;
tee STDOUT, '>>&', $LOGFILE or die $!;
...
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update: I am thinking that this is not going to work properly either, because the tee work is done by a subprocess and so there is a race condition.

In this case, IO::Tee that uses a tied file handle to duplicate the output may be a better option.