Contacts / Connection which way to go

I am trying to figure out the best way to relate contacts to multiple companies. The issue I am having is you can only relate a contact to a single primary account. I would like to be able to have a contact and relate them to multiple companies. I have a contact who is an IT vendor who handles the IT services for multiple accounts. When I try to add the contact to more than one account it fails saying a contact can only have 1 primary account.

Connections

I am looking at using connection to relate the contacts to multiple accounts however when I go to select a primary contact in a service activity or case it will not sort it under the contact view/related contacts.

Anyone have any ideas on the best way to accomplish this?