“The chance that the patch in a ring contains the same species (to continue the chain) depends on the ring. Starting from the closest ring, the chances are 28%, 48%, 68%, and 88%. Catching a Pokémon rather than defeating it will increase these odds by 10% for the patches automatically generated after the battle.”
I believe you misread. It doesn’t increase the shiny odds, it increases the odds of the generated patches being the same species. Also, it only increases the odds for the generated patches after the battle, so it doesn’t stack.
Source: What else? Bulbapedia