Custom Search without "s" variable

I’ve created a custom search for my blog, which searches custom post types by all sorts of post_meta values.
I’ve got the actual search working great, but I was building it on a homepage to test it, so the form submits to the same page and the loop of results is also displayed there. Everything working.

But now to implement it, I want the results to appear on a new page (I suppose a custom version of ‘search.php’) but I’m confused and can’t get it to work.

I was using a search form like this (this is without all the extra <input> sections):

<form name="search" id="searchform" action="<?php bloginfo('url'); ?>/" method="get" role="search">
<input type="text" class="field" name="keyword" id="keyword" placeholder="<?php esc_attr_e( 'Enter to search &rarr;', 'agility' ); ?>" value="<?php if(isset( $_GET['keyword'] ) ){ echo $keyword; }?>"/>
<input type="submit" class="submit" name="submit" id="searchsubmit" value="Search" />

However, this fails to send everything to search.php. I figured out that for some reason WordPress needs to have the “s” variable or it doesn’t work. If I was to change the second line to:

<input type="text" class="field" name="s" id="s" placeholder="<?php esc_attr_e( 'Enter to search &rarr;', 'agility' ); ?>" value="<?php if(isset( $_GET['keyword'] ) ){ echo $keyword; }?>"/>

Then it works fine. The problem is, I want people to be able to search without having to put anything in the keyword field. For example, they might choose from a drop-down to search everything located in the UK, but the search form submits with “s=” (no value for ‘s’) and so WordPress doesn’t go to search.php at all, instead just returning a standard blog loop page.

Is there some way to get around this?

All I want is to be able to submit my search to a page that can handle the custom search.

Any ideas?

Thanks, and hope this makes sense.