For Magento 1.9.x.x, you have at least two options:
1) Update the default
contacts.phtml file located at
\app\design\frontend\base\default\template\contacts\form.phtml (if you have a custom theme, use that directory - or copy this file to your custom theme in the same directory as to not override the default)
From there, you can use this file and basic
form structure to change the fields that are available.
You can check which
.phtml file is loading by using the
template path hints option in your
System > Configuration > (change to store view instead of global) > Advanced > Developer > (include your IP in the field at top) > set Template Path Hints to YES > Save Config - then reload your
/contacts URL and you should see a path to which file is being loaded.
2) Use MageMe's WebForms extension which contains a better GUI based form builder that allows you to override the default
There are other "contact form" extensions that alter how this page looks, but in my experience, webforms has been a lifesaver - and has utility beyond just the