You don't have variables named $h3 or $h4. You probably mean %h3 and %h4.

You don't change variable %h3. You change an element of %h1 using a reference stored in %h2 and another reference stored in %h3.

my %h1 = ( k1 => "v1"  );
my %h2 = ( k2 => \%h1  );
my %h3 = ( k3 => \%h2  );

         %h3
+--------------------+
|       +----------+ |
| k3 => | Ref ------------+
|       +----------+ |    |
+--------------------+    |
                          |
                          v
                         %h2
                +--------------------+
                |       +----------+ |
                | k2 => | Ref -----------+
                |       +----------+ |   |
                +--------------------+   |
                                         |
                                         v
                                        %h1
                               +--------------------+
                               |       +----------+ |
                               | k1 => | Str "v1" | |
                               |       +----------+ |
                               +--------------------+
[download]

my %h4;

         %h3
+--------------------+
|       +----------+ |
| k3 => | Ref ------------+
|       +----------+ |    |
+--------------------+    |
                          |
         %h4              |
+--------------------+    |
|                    |    |
|                    |    |
|                    |    |
+--------------------+    |
                          |
                          v
                         %h2
                +--------------------+
                |       +----------+ |
                | k2 => | Ref -----------+
                |       +----------+ |   |
                +--------------------+   |
                                         |
                                         v
                                        %h1
                               +--------------------+
                               |       +----------+ |
                               | k1 => | Str "v1" | |
                               |       +----------+ |
                               +--------------------+
[download]

%h4 = %h3;

         %h3
+--------------------+
|       +----------+ |
| k3 => | Ref -------------+
|       +----------+ |     |
+--------------------+     |
                           |
         %h4               |
+--------------------+     |
|       +----------+ |     |
| k3 => | Ref -----------+ |
|       +----------+ |   | |
+--------------------+   | |
                         | |
                         v v
                         %h2
                +--------------------+
                |       +----------+ |
                | k2 => | Ref -----------+
                |       +----------+ |   |
                +--------------------+   |
                                         |
                                         v
                                        %h1
                               +--------------------+
                               |       +----------+ |
                               | k1 => | Str "v1" | |
                               |       +----------+ |
                               +--------------------+
[download]

$h3{k3}{k2}{k1} = "v2";

         %h3
+--------------------+
|       +----------+ |
| k3 => | Ref -------------+
|       +----------+ |     |
+--------------------+     |
                           |
         %h4               |
+--------------------+     |
|       +----------+ |     |
| k3 => | Ref -----------+ |
|       +----------+ |   | |
+--------------------+   | |
                         | |
                         v v
                         %h2
                +--------------------+
                |       +----------+ |
                | k2 => | Ref -----------+
                |       +----------+ |   |
                +--------------------+   |
                                         |
                                         v
                                        %h1
                               +--------------------+
                               |       +----------+ |
                               | k1 => | Str "v2" | |
                               |       +----------+ |
                               +--------------------+
[download]

You can use Storable's dclone to do a "deep" (recursive) copy, in which case you'll end up with the following:

         %h3
+--------------------+
|       +----------+ |
| k3 => | Ref ------------+
|       +----------+ |    |
+--------------------+    |
                          |
                          v
                         %h2
                +--------------------+
                |       +----------+ |
                | k2 => | Ref -----------+
                |       +----------+ |   |
                +--------------------+   |
                                         |
                                         v
                                        %h1
                               +--------------------+
                               |       +----------+ |
                               | k1 => | Str "v2" | |
                               |       +----------+ |
                               +--------------------+

         %h4
+--------------------+
|       +----------+ |
| k3 => | Ref ------------+
|       +----------+ |    |
+--------------------+    |
                          |
                          v
                        %anon
                +--------------------+
                |       +----------+ |
                | k2 => | Ref -----------+
                |       +----------+ |   |
                +--------------------+   |
                                         |
                                         v
                                       %anon
                               +--------------------+
                               |       +----------+ |
                               | k1 => | Str "v1" | |
                               |       +----------+ |
                               +--------------------+
[download]