While Apex tests are very much a key tool in monitoring regressions in your application, the limitations and scope of functionality they can test is small, particularly to perform volume or user interface testing (clients using JavaScript for example).
Note
You might wonder why the Ant script used by Jenkins deploys twice, as the check deploy to the packaging org will confirm whether the Apex code compiles and that all Apex tests complete successfully. The reason is that having the build org deployed with the latest build allows for it go on to be used for other purposes, such as executing further regression testing steps.
Here are some considerations to implement further testing approaches: