No need to start a new project. Just open it in Xcode and it will be upgraded. Make sure you compile against the latest Base SDK, while allowing a minimum deployment target of what you need (iOS6/iOS6.1). You can use the new preview assistant view in interface builder to simulate how it would look in iOS6 and iOS7. You can set deltas in interface builder to set different values between iOS7 and iOS6.
You can download the iOS6.1 simulator from Xcode 5 settings.