URL for Contact info search criteria

 I have built an external web site that connects to CRM to search for contacts using their phone numbers. The URL we pass in works properly and will return the Row ID of the contact(s) that have the phone number in their record. If there are multiple contacts sharing the same phone number, multiple Row IDs are returned. We then use another URL to open CRM to the Contact page of the first Row ID returned. This is okay for a single match, but not good for multiple matches.

What I would like to do instead is pass the phone number in via a URL and have CRM open the search results page as it would do if you entered the phone number in the action bar search. Looking at the URL when you do the search from the action bar, I don't see where the phone number is passed in for the search. This is the search URL from the action bar: