Most missions are fine, but there are some that just shower us with gear - like some VC supreme missions, mech missions (like the one in the Ecru VC) and a few others. Even with no +loot quantity gear, the amount of drops is crazy and the vast majority of them have to be discarded item by item if you want to continue picking up more stuff. I just did the mech Ecru mission and had to discard gear that would fill my inventory three times over, easily. 

Why do you design missions where most of the gear must be destroyed or not picked up? What's the point of wasting so much time deleting items manually if you want to make space for better loot? Do you think it is good gameplay? Is it fun?

Adding new tabs (just like you did with our storage space) would help, but I don't think it would solve the problem entirely. But there could be other ways to do that:

• Add an ability to clear the inventory during the mission - possibly by converting the stuff into credits by using a limited-use stackable consumable (could be sold by custodian Rebus, or as part of the Sacred Vault rotation, for example).
• Place hidden cogitators inside missions that would allow us to do the same (convert gear in our inventory).
• Edit: if you don't want to allow conversion to credits for lore / immersion / whatever reason, we could have an option to stack the dropped gear into caches. I.e. the reverse process of opening a reward cache. The game now allows up to 8 items per cache - 'compressing' dropped gear at this 1:8 rate, to be opened once back on the ship would help immensely.
• Re-design the way loot pick up works. I don't think there is any real system in place, the order seems to be either random or first dropped - first collected. So you end up auto picking up all kinds of blue crap before you actually collect that shiny ancient relic glittering at your feet. There should be a system that auto picks up loot in the descending order based on item rarity / quality.

The last one is especially important when your inventory is full. Take a look at this screenshot:

What do you think will get auto picked up first? The relic grenade launcher or the 20 useless blue and green drops? And when your inventory is full, how will you actually get that relic drop? Except going into your inventory and manually deleting random gear, which keeps getting replaced by that blue and green junk you see on the ground and which you have to keep deleting item by item before the relic is finally picked up. The other option is to switch looting to manual mode, which is a bother and you either have to manually pick them all up or lose them (thus losing crafting mats and credits)...