'Search Form' block I would like to add a link called "Advanced Search" below the existing Search box Within our 'Search Form' block I would like to add a link called "Advanced Search" below the existing Search box.  One possible way of implementing this would be to utilize hook_block_view_alter or hook_block_view_MODULE_DELTA_alter within a custom module and then add this new link to the bottom of this block.
Is this the best way to add this link to the 'Search Form' block?
You could either use the hook_block_view_alter to add the link, just like you say. Or you can implement the search-block-form.tpl.php in your theme and add the link in the template file.