You can't so much override a function as replace it.

use Module;  # Execute module first so it doesn't replace you!

BEGIN {
   package Module;
   no warnings 'override';
   sub func {
      ...
   }
}
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Alternate syntax:

use Module;  # Execute module first so it doesn't replace you!

BEGIN {
   package Module;
   no warnings 'override';
   *func = sub {
      ...
   };
}
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You can wrap the original function:

use Module;  # Execute module first so it doesn't replace you!

BEGIN {
   package Module;
   no warnings 'override';
   my $orig_func = \&func;
   sub func {
      ...
      $orig_func->(...)
      ...
   };
}
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You can even do so temporarily:

use Module;  # Execute module first so it doesn't replace you!

{
   package Module;
   no warnings 'override';
   my $orig_func = \&func;
   local *func = sub {
      ...
      $orig_func->(...)
      ...
   };
   # Replaced from here.
   ...
   # Replaced until here.
}
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In all cases, it's replaced for everyone. You could use caller to emulate other behaviours, perhaps.

The only weirdness I can find in this module is that the absUrl subroutine is only called in MIME::Lite::HTML itself in private subroutines of the, "parse" method, like this:

There's no such thing as a private subroutine. The inner sub is just as public as the outer one. Whoever wrote that code was fooling himself. And he's asking for trouble. The following warning is an understated indicator of the problem of nesting named subs:

>perl -c -we"sub foo { my $x; sub bar { $x } }"
Variable "$x" will not stay shared at -e line 1.
-e syntax OK
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I don't know if I've answered your question since I really don't know what you're asking, but I hope I helped.

Note: The BEGIN is not necessarily needed in any of the examples. Plain curlies would likely work too.