In case you have a reasonably recent version of HTML::Form (being used under the hood) - i.e. one which has the strict method - you should be able to add new fields to the form simply by setting them (strict is off by default) — meaning you won't get "No such field..." errors (as in older versions before v5.817 / 2008-10-10):

...
$mech->field('password','welcome');
$mech->field('rnd','...some random number');   #  <-- possibly new
$mech->click();
[download]

The additional 'rnd' field/parameter will get submitted just as the original ones.