By trying to redeem evil people who you see may have a chance at redemption like a Paladin might instead of just killing them outright.
No, a Paladin does no redeeming himself. A Paladin is only expected to spare people that are
displaying honest intent to turn their life around. If there is even an inch of doubt or dishonesty present, a Paladin might actually allow evil to fester a few years down the road by letting them go. What you are talking about is being 'lawful stupid' instead of 'lawful good', Paladins aren't choirboys.
Even though I'm not the biggest fan of 'Wrath of the Righteous', it does contain one of my most favorite conversations in any RPG's. After a local congregation of Shelyn (goddess of art, love, etc...) is wiped out by cultists, you can find the leader of the cultists still at the place of slaughter. One of your party members is a cleric of Shelyn and, once you've cornered him, wants to execute the cultist leader. However this cultist mockingly states that he wants to redeem himself, and that Shelyn's teaching dictate that, no matter how small the chance, the goddess requires the cleric to let him repent so that he might create beautiful art. The cleric is livid and has a crisis of faith, but as a Paladin you can comfort him, causing him to not steer away from his faith and forgiving the cultist leader. If you're able to persuade the cleric to forgive the cultist leader, you get the dialogue option as a Paladin to state 'But I'm not of Shelyn, so in the name of my god I sentence you to death, executed by my hand. Immediately'.
Yes, you have 'Paladins of Shelyn', but both Pathfinder games make it a point to show how useless they are. The personification of 'lawful stupid'.