First of all, I'm glad you removed the "Cozy/Snuggle/Happy Hut" anyway, because they regularly KILL pet parrots, specifically Conures...Hundreds if not thousands of pet parrots of all species have been killed by those "Huts", to the point where there is a Class-Action Lawsuit against the 2 main manufacturers of them, and they started printing "Not intended for use with Conures" on the front of them because so many different pet Conures of all species have died directly as a result of them being in their cages...They either slowly pick tiny pieces of fabric off of the inside over-time and swallow it, and it's not digestible, so it accumulates inside of their stomachs over months, until you find them dead in the bottom of the cage one day...OR one of the many threads that are inside of them come loose one day when you're not home and your bird either gets it's neck caught in it and hangs himself, or he gets his leg caught in it and hangs from it all day long, and either dies of the blood-loss or loses his leg...And in addition to the danger of the Huts is the fact that ANY small, dark places you provide to your bird and allow him to go underneath or inside of, like a Hut, Tent, Hammock, Beds, or any type of Boxes, or underneath furniture, behind pillows, underneath blankets, towels, etc. will cause their sex-hormones to go nuts and cause everything from severe aggression in males to chronic egg-laying in females. So no more Huts or anything of the like anywhere, they sleep just fine on a perch.
***Honestly, and I'm just being honest and giving my professional and personal opinion here, even if the size of the cage isn't the reason he doesn't like being inside of it (and I think that it 100% is), that cage is WAY TOO SMALL for a Green Cheek Conure...My Budgies stay in cages that size, one Budgie at a time...That cage would be too small for a Cockatiel for sure, and way too small for a Conure of any species...So either way I'd look into getting him a much larger cage, it needs to be wider and longer at the least, and taller would be better...