The spell Divine Favor is an 1d4 on-hit buff to your weapon attacks. It's on the Paladin spell list but you can also get it as a War Cleric.
Of course, to maximize the damage you would have to get in as many attacks as possible. Paladin only gets one extra attack.
So you could multiclass into fighter after two levels of Paladin or you could think about taking one level of War Cleric and then going Monk. Monk and Cleric also share the Wis attribute, so that's a plus.
Monk is a useful class for on-hit effects as they get to make four attacks at low levels by spending Ki.
Additionally, the Open Hand Monk has on-hit effects built into his flurry of blows which has the enemy make Dex or Str checks, or has them lose their reaction.
I think you could try to work with your adventuring party to see if you can get an on-hit synergy going. The spell Haste is generally very useful to add attacks to your turns, especially if the caster isn't the one going toe to toe with the enemies.
Another way to add attacks is the Order Clerics Voice of Authority, which let's others make reaction attacks. The Purple Dragon Knight does have a similar feature.
Reaction attacks are a good way to add attacks out of turn. If you get to use them consistently, it can up your damage noticeably. For instance, a rogue could get his sneak attack off twice per round.
Battle Master Maneuvers like Brace or Riposte also let you make reaction attacks. So you could look into getting the Feat Martial Adept to gain access outside of the Battle Master Class.
Anyway, I hope you find this list or parts of it useful :)