A mission type like the old covert operative exalt missions would be pretty cool I think.

Maybe it could be stealth focused with you limited in number of soldiers and favoring sneaking past as many enemies as you could to the objective.

Another idea would be a chryssalid hive attack like the whaling ship mission in the previous game. That was always an intense mission. Something like that would be awesome. Give even more of the starship troopers vibe than usual.

Perhaps a defense against huge waves of chryssalids while you hold a fortified position with some xcom turrets to help you. You have to hold a certain number of turns while dozens of chryssalids assault you.

Hack multiple access points on the map. No time limit but everytime you hack, there's a reinforcement coming on your location + nearby pod is alerted

I just thought of a cool alternative mission type:

At X-COM, word of a public resistance fighter execution comes through the comms - within the next 2 days (get to the mission site before 48 hours), Advent will be publicly executing a resistance VIP.

When the X-COM squad drops in, they have 8 turns (or however many) to get to the VIP and take out the executioners. If the time runs out and the executing pod hasn't been engaged, it'll shoot the VIP (note that in early missions this means the VIP *might* survive the first shot if time runs out). If the executing pod HAS been engaged, control of the VIP is given to the player, allowing him/her to run, but the pod will still prioritize the VIP.

ADVENT figure they can take care of XCOM, so they don't call in reinforcements until after the VIP is secure. Once the VIP is secure, an evac zone is designated and the squad has to reach it before an additional 6-8 turns expire and ADVENT's retaliation prevents Firebrand from extracting any remaining soldiers.

Because the execution is public, it would always occur either in a large building or outside (in a parking lot, on a building's facade, etc.).

The Objectives, then, would be:
*Rescue VIP before the public ADVENT execution
*Extract the VIP at designated location
*Extract all XCOM soldiers

Perhaps a simple variation on the enemy VIP mission where you are authorized to use lethal force on said VIP, and you still get the full prize. This is offset by additional enemies than the vanilla dark VIP mission type.

A squad of traitors.
A squad of soldiers advent defected to the resistance. Its destruction sent numerous forces. The XCOM soldiers together with a group of advent have to repel the attack.

Reward for completing the mission:
The ability to use a squad of soldiers of the advent in some missions. (friendly reinforcements can be)
Intel.