Hello PokeFanForever,
If you are hunting legendaries, soft resetting is the only way to do so (unless you have US/UM, in which you can use Ultra Wormholes to hunt certain legendaries, post-game). The only way to increase those odds is to use the Shiny Charm, but chances are that you do not own that yet.
If you are not hunting legendaries or searching for wild shinies, then the Masuda method mentioned above works wonders.
If you really want to search for wild Pokemon (I would only recommend doing this if the Pokemon is genderless or they are legendary/mythical/ultra beast, otherwise use the Masuda method), then you need the move False Swipe. That leaves the opposing Pokemon with at least 1HP, so your catching chances are greatly increased.
Another great Pokemon for this is Tapu Koko with the move Nature's Madness. Tapunium Z boosts this attack so that it removes 75% of the opposing Pokemon's HP, so it is better used for higher level/legendary Pokemon before using False Swipe (Tapu Koko can also learn False Swipe).
Heal Pulse can help as well. If legendaries are running low on PP for their moves, they will soon use Struggle, which will damage themselves. Heal Pulse can prevent an accidental shiny death by healing them, but it is only helpful in rare cases.
For most Pokemon, I would use Vom's method, but for certain Pokemon the above information should help with your problem.
Good luck, and may the shiny odds be ever in your favour!