This question has been answered by treigh. See the first response.
<form id="my_id" action="[[~[[+landing:default=`[[*id]]`]]]]" method="[[+method:default=`get`]]">
// Reference the name of this search input field in your SimpleSearch snippet call.
<input id="searchField" class="my_class" type="text" name="my_custom_search_field" value="[[+searchValue:default=`Search the site`]]">
<input id="searchIcon" class="utilityButton" type="image" alt="Search" src="/assets/templates/my/images/searchButton.png">
<input type="hidden" name="id" value="[[+landing:default=[[*id]]]]">
</form>[[!SimpleSearch? &searchIndex=`my_custom_search_field`]]