Well, you're already pretty tanky, but there's always more you can do.
Short-term: If your splint armour is not magical you should definitely try to upgrade to plate if you can. It doesn't cost anything but gold, and unless your DM allows you to buy magic items there's just not that much use for gold. You can try to persuade your party members to help you out here, since they have even less things to spend gold on.
Also in the short-term, while your AC is already huge, you can still improve it. There's a 1st-level Paladin spell called Shield of Faith which is worth casting if you think you're getting into a tough fight. It raises your AC by 2 as long as you can maintain concentration.
Finally, in short-term solutions, think about how often you've been using your Protection Fighting Style. If you've been using it a lot, great! If not, it might be worth asking the DM if you can change it to the Defense Fighting Style. "I've hardly ever used it" is an argument I think most DMs will be OK with, so it's worth thinking about.
Level 7: You should definitely not multiclass at this level, since you're about to get resistance to all damage from spells. This is just about as tanky as it's possible to be as far as spells are concerned. You don't get much else at this level but this feature is huge.
Level 8: You probably don't want to multiclass at this level either. Ability score increases/feats are really important, and you won't get one if you mlticlass. The big decision here becomes what you should do with your ability score increase/feat. You've got a lot of options.
Upgrading your Strength increases your damage, which doesn't directly make you tankier, but killing things completely stops them from damaging you, so in a sense it's the best way to tank.
Upgrading your Charisma boosts all your saving throws, which will make you a bit tankier. It's also nice for your allies, and any spells you use that involve the enemy making a saving throw.
Upgrading your Constitution gives you an extra 8 hp and makes it easier to maintain concentration. Definitely a solid option, but nothing flashy.
For feats, there are even more options. Heavy Armour Master is a great feat at low levels, but at level 8 it's not looking that good. 3 damage is not a large amount at this point, and more importantly, less and less of the damage you're tanking is non-magical.
Tough is a straightforward +16 to your HP - worth considering, but it's better to avoid damage than take more of it, and you should have around 68 already.
Resilient and War Caster both make it easier to maintain concentration, but don't do much else for you. If your Constitution was an odd number I'd recommend Resilient for sure, but as it is I'm not sure it's worth it for you.
Level 9: It's definitely worth thinking about multiclassing - you've got most of the Paladin's features and from here on out they mostly improve rather than getting new ones.
So, staying with Paladin - definitely still a good option. There are some nice goodies coming up, and your spells will steadily improve, so don't dismiss it just because there's nothing as big as the feature you got at level 7. That's just because that's such a good feature, not because the other features are bad.
Barbarian: Probably not. Rage is nice but it's tricky when you have spells to consider, and Unarmoured Defense just doesn't work for you.
Bard or Sorcerer: compatible stats-wise, but just don't offer you that much beyond more spell slots to smite with.
Cleric, Druid, or Wizard: Incompatible stats and doesn't really offer you anything except more spell slots.
Fighter: Another Fighting Style, which is nice, especially if you can't or don't want to talk your DM into letting you change the one you've got. Action Surge at level 2 is great as well, after that it tapers off in usefulness as far as you're concerned.
Monk: No. Nothing here is good for you.
Ranger: Another Fighting Style at level 2, but if you want that, it's probably better to get it from Fighter.
Rogue: There's not much here for you without serious investment. 3 levels can get you Assassinate, which vastly increases your damage if you can set it up, but it doesn't make you any tankier till level 5.
Warlock: Definitely saving the best till last. 3 levels of Warlock gets you the Darkness + Devil's Sight combo, which effectively gives you advantage all the time and your enemies disadvantage all the time. 3 levels is a serious investment, though, so think carefully before you do this.
So overall, consider 1 or 2 levels of Fighter (or 4 to maintain ability score increases), 3 levels of Warlock (or 4 to maintain ability score increases, or just sticking it out as a Paladin.
