What's the key to a strong relationship with your software outsourcing company? I believe the key is communication. That's why at WorkingMouse we have the Partner Journey.
The Partner Journey is a step-by-step guide to the whole partner process, starting with the Brief stage and continuing post-development into the Support stage. It spells out everyone's role and responsibilities, and ensures everyone is on the same page, every step of the way.
Some of these considerations are related. For example, a local developer can be more available and have better communication regimes than one that's located offshore in India.
Actually meeting the developers, designers, and testers that will be working on your application, is important, and it sure beats Skype.
Another pairing is price and post-development support. A developer that doesn't offer post-development support could be planning on rushing through the development stage and releasing buggy software. Cheapest isn't always most cost-effective.
Check that you won't have the same problems with your new, potential developer as you did with you old one.
Once you have signed a new developer, it's time to turn your attention back to your existing one.