Create an inherited view of the res.partner.select view and add one line per field that you like to search. For example:

<?xml version="1.0"?>
<filter string="Country" position="after">
<filter string="City" context="{'group_by': 'city'}"/>
<field name="street" filter_domain="[('street','ilike',self)]"/>
<field name="zip" filter_domain="[('zip','ilike',self)]"/>
<field name="city" filter_domain="[('city','ilike',self)]"/>
<field name="comment" filter_domain="[('comment','ilike',self)]"/>
</filter>