Some side handles won't fit the Tilta cage, but some will.

Something like the SmallRig 2222 could work since it only needs one thread in the bottom of the cage, but isn't ideal for ergonomics or the added weight, and is slightly expensive.

What I did when I had the Tilta cage that worked really well was to put this NATO rail permanently on the side of the cage, and then got this cheap NATO side handle. Worked like a charm and was much more versatile than screwing something in since you could quickly adjust the height and take it on/off. The 7cm NATO rail was the perfect size for the Tilta cage.

These however will NOT work: handles that have 2 mounting screws (like the Smallrig 2093 or this Camvate one). So don't get those with Tilta.

EDIT: Also Tilta does make a side handle, but it's not widely available yet to my knowledge. And it looks pretty cool honestly, but it falls victim to all the above issues. It uses such a unique mounting system that they'll probably never use that design again.