You shouldn't start a new game. You've already got most of the resources for enhancing your Pokemon's stats open to you now. Why get rid of it? All you need to do is just breed new Pokemon with the right natures
The benefits of breeding a Pokemon even though you may already have the right nature is IVs. IVs, also known as Individual Values, are hidden values between 0 and 31 that determine how high a Pokemon's stats can get. People usually breed because their Pokemon may not have very high IVs.
To know which Pokemon give which stat, I like to use
this.
Sorry if this doesn't make sense. ;^^