Have you tried it like this:

while (< $$self{fhFrom} >) {
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UPDATE: Sorry, I should have tried that myself before posting it. It doesn't seem to work as hoped for:

perl -e 'open(my $fh,".bashrc");$h={ f => $fh };while(<$$h{f}>){print}
+'
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that just prints "GLOB(0x00blah)", which is not what I wanted.

Assigning the hashref value to a temporary scalar variable seems to be the only way to make the diamond operator do what you expect it to do. I think I've seen an explanation for this somewhere...

ANOTHER UPDATE: It seems that it is not the dereference ($$ref{key} or $ref->{key}) that causes the problem -- it's just the presence of anything other than a plain scalar variable:

$ perl -MData::Dumper=Dumper -e 'open(my $fh,".bashrc") or die;
 my %h=(f => $fh);print Dumper(\%h); while(<$h{f}>){print}'

$VAR1 = {
          'f' => \*{'::$fh'}
        };
GLOB(0x800d80)

$ perl -MData::Dumper=Dumper -e 'open(my $fh,".bashrc") or die;
 my @h=($fh);print Dumper(\@h); while(<$h[0]>){print}'

$VAR1 = [
          \*{'::$fh'}
        ];
GLOB(0x800d80)
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(i.e. both cases fail to actually read from the file handle; the while loop prints the GLOB thing instead)