You can actually give explicit package name to satisfy strict (like it says). Like this:

%main::GLOBAL = (
    dir1 = "/path/to/directory",
    dir2 = "/path/to/another",
);
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and then in read_config_files:

sub read_config_files {
    $main::GLOBAL{newstuff} = "a string that was added";
    # etc
}
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That's assuming that %GLOBAL lives in the main package, which I suppose it does. For refactoring, perhaps it would be clearer to give it its own package:

# this is file global.pl

use strict;
use warnings;

package GlobalStuff;

%GlobalStuff::GLOBAL = (
    dir1 = "/path/to/directory",
    dir2 = "/path/to/another",
);

# or, alternatively (same thing)
our %GLOBAL = (
    dir1 = "/path/to/directory",
    dir2 = "/path/to/another",
);
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and

# this is file config.pl

use strict;
use warnings;

require 'global.pl';

read_GLOBAL();
print_GLOBAL();  # for example

sub read_GLOBAL {
    $GlobalStuff::GLOBAL{new} = 'a string';
}

sub print_GLOBAL {
    while ( my ($k, $val) = each %GlobalStuff::GLOBAL ) {
        print "$k => $val\n";
    }
}
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Then I would rename 'global.pl' to 'GlobalStuff.pm', and change all calls to require accordingly:

require GlobalStuff;
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require appends .pm to a bareword and searches this file in @INC; see require for details.