Renunciation of material things, and bringing more contentment usually comes about from having more Unselfish Emotions in our practice, and gaining contentment from the development of inner qualities, then we have much more inner spiritual happiness.
This is what the Buddha teaches us when he says: "In order to let go of worldly happiness, one cultivates spiritual happiness". For lay people it's quite difficult to get contentment from high stages of concentration, because our lifestyles don't usually support that, but we have wonderful opportunities to develop the Brahma Viharas - the unselfish emotions - because we are in contact with people most of the time. We can develop these unselfish emotions, especially doing more generosity work, to bring more contentment with ourselves.
We have to know what brings spiritual happiness. For some people it comes about through a feeling of refuge. The note goes on to say, "chanting", this can bring about a feeling of refuge giving us a feeling of inner contentment; so that we don't need to seek happiness externally in material things, so much. We can find a feeling of confidence within ourselves.
Mindfulness of the present moment brings more contentment, because usually we have a lot of desire when we're living in the future, and wanting something from the senses in the future. So by becoming more aware in the present moment, we start to see what we already have; what is already present.
Regarding "Sport" - having a healthy body is very helpful, so - yes, being mindful when we do sport or walk in nature, are very useful, calming activities, to find more inner happiness.
However, from my experience, the main things that helps to bring about Renunciation are the Compassionate Intention and the Unselfish Emotions. These bring about an emotional maturity that supports any wisdom practice. If we haven't got that emotional maturity, the feeling that we have to "give up" something, is too forced, it doesn't happen naturally. So, developing this Compassionate Intention; opening out to the Dukkha of existence; learning how to feel with it; developing Lovingkindness - the friendliness towards ourselves and other living beings; developing Joy with our own and others' good qualities. These types of qualities bring more inner happiness and we will often see that it's superior to the happiness we used to seek, because it depends on our own inner qualities, and not on others.