Based on the documentation, it appears there is no way to simultaneously execute an email service and the email to case standard service.
When defining email services, note the following: An email service only processes messages it receives at one of its addresses.
Am I missing something or is there some other way to invoke email-to-case and an apex email service at the same time? I would like to create contacts for every inbound email to an email to case email and thought this would be best practice. The documentation example, led me to believe you would want to create task on contacts to signify there is something to action within SF, not within an external system. https://developer.salesforce.com/docs/atlas.en-us.apexcode.meta/apexcode/apex_classes_email_inbound_using.htm
If it is not best practice or not an option, I believe my next step would be to utilize a trigger or perhaps a process builder on only during case insert for creating contacts on after inserts of the email-to-case case to extract name and email information from the inbound email.