packages can't surround each other, your packages Test and Page are on the same level.

I know that packages don't nest, and that namespaces are not organized in a hierarchy. What I am after is information about when package switching occurs, and after switching, whence. This information must be known to the compiler to restore the previous package's scope after the current package's scope ends:

package Foo {
   ...
   package Bar {
      ... # I want to know 'whence' here
   };
   ... # package Foo resumed.
   ... # the compiler knows about the package to be resumed.
   ... # how can I use that information inside 'Bar'?
};
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I don't think so. Yes, it's a partial quote, but what I left off (end of surrounding block etc.) doesn't change much. When a package begins, the previous package ends. As I understand it, at end of enclosing block, the package ends, and you are in package main.
Update: I was completely wrong, see below...

As I understand it, at end of enclosing block, the package ends, and you are in package main.

No. You are back in the package you came from:

package Foo {
    print "(1) package " . __PACKAGE__ . $/;
    package Bar {
        print "(2) package " . __PACKAGE__ . $/;
    }
    print "(3) package " . __PACKAGE__ . $/;
}
print "(4) package " . __PACKAGE__ . $/;
__END__
(1) package Foo
(2) package Bar
(3) package Foo
(4) package main
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perl -le'print map{pack c,($-++?1:13)+ord}split//,ESEL'