Here’s a few I’ve used that have worked well:

• Three Sided Battle: The party encounters two different groups of NPCs fighting each other. They can join up with one side, but there will be consequences if either side is left undefeated.
• Weak Spot: The enemy can only be damaged by attacking it from a certain direction or by targeting a certain part of it.
• Two Health Bars: An enemy has two forms that it can swap between at will. Each form has a distinct health bar and if one of the forms is defeated, the enemy is locked into the other form which then becomes more powerful.
• Mirror Match. Is there a really powerful ability a PC has that everyone has been enjoying on their side for the whole campaign? Give it to an enemy.
• Tactical Enemies: Certain formations give the enemies massive bonuses so the PCs need to break apart the enemy formations.
• Hiding Spots: There is an area where the enemies are safe from harm and the players must figure out a way to draw the enemies out.
• Death Zone: There is an area highly dangerous to the players, such as lava or acid, that doesn’t harm the enemies, and the enemies’ strategy is to drag the players into the death zone. Can combine with Hiding Spots as well.

Alternate win conditions is my favourite way to spice up combat, although I think it's important to drop a healthy amount of normal fights to the death in, because many characters are only at their best when outputting damage.

On top of that, considering that if you make the goal to calm a rampaging giant, the bard with a suite of enchantment spells may find that trivial while the barbarian may have nothing on their sheet that can aid them. One solution is to introduce interactive objects such as a voice amplifing horn on a roof to speak to the giant, but that still favours mental and social skills, which will favour spellcasters in the end.

My piece of advice which I'll swear by is inelegant but has always worked for me; ensure there is something dynamic in the encounter that dealing damage to is good. Perhaps it's knocking down a monument to block a road and cause the giant to steer (in terms of telegraphing something like this, just tell your players it's an option, you don't need to be cryptic), but the monument is guarded, making an ettwmot to knock it over more dynamic. Better yet, make the giant unable to be calmed because a band of unselie pixies and quicklings are slashing at his back and legs, and those quicklings work for a fey antagonist to the game. Suddenly the characters who are built to be dynamic can stil ltry to charm, convince and steer the giant, but the PCs made to deal damage can simply battle these fey. Assume the party are reluctant to kill the giant because they need information from him, or he's a friend.

The distilled version of this idea is this: "the PCs have aim A, dealing damage will either not help them achieve it or make their lives more difficult once complete. An antagonist force is causing the stakes of aim A to rise, so aim B is to thwart them, and in this case, damaging them is fine. That's a format that is vogue enough that using it multiple times would often go unnoticed.

One other suggestion is a little more complex; have a series of dynamic encounters all related to the same goal. This goal can be lifted from a game such as capture the flag. Suddenly splitting the party is fine as your paladin may stay in one room to guard your flag (or totem or whatever) while your hasted monk dashes ahead to capture your opponents. Alternatively, it's a bomb planting goal. An arcane explosive must be planted in a room to disable to infernal portal, but each point is guarded by cultists. As your party arrive in room A to take control of it and plant the bomb, the cultists in room B rotate round behind you, and now you must defend room A. Perhaps you give the arcane bomb to your rogue and tell them to hide outside room A until the party attack B, and when the cultists rotate leaving A unguarded for the rogue to sneak in and plant the device.

This latter option is always fun to do but it's really important to consider that if your party don't typically have many encounters or don't often delve into dungeons they'll come up much less often, and that's fine.

My final piece of advice is simply to ensure that you have at least two kinds of enemy that function very differently, which could be as simple as archers and knights. You will always have dynamic encounters because no battle will ever play out the same.