The normal approach, irrespective of scope, is to have a single copy of the string and pass scalar references around. For example:

my $bigref = bigget;
... do something ...
do_something_more( $bigref );

sub bigget {
    local $/ = undef(); # disable carriage control locally
    open my $fh, $ENV{BIG_STUFF} or die "$!: \$ENV{BIG_STUFF}\n";
    my $slurp = <$fh>;
    close $fh;
    \$slurp; # return the reference to $slurp, which also keeps it ali
+ve outside this scope!!!
} 

sub do_something_more {
    my $bigref = shift; # copying the ref not the string
    $$bigref =~ /^\s+(\S+)/; # using a dereference
    my $first_word = $1;
    ... do more ...
}
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