For future reference and easy access, I'm posting a transition into deep half guard I want to work on:
You have half guard on their left leg. Your right hand is ready to defend the crossface.
Your left arm is underhooking their right leg. Ideally, this would be really deep, with their leg on your shoulder, but realistically you’ll just get a hand under their knee.
When they go for the crossface, duck your head and swim your right hand under their armpit.
Shove their armpit and ribs to send their upper body away.
The key is getting my hips underneath their hips. From there I can lift them and break their base. I really need to commit to the movement of switching and throwing my hips under them. Keep scooting, shrimping, bumping under like I want to go out the back door.
The traffic cop hand shoves their armpit to throw their weight forward (which helps get me under their hips). They also can't crossface since they can't turn back towards me and they need to use their arms to post or they'll face plant.
Once I'm deep enough, I can switch to a gable grip so I can pull their leg on to my shoulder to spread their base and keep them leaning forward. Like in the electric chair sweep:
I can also keep the normal underhook and switch back to the traditional half guard if they base back and won't let me go out the back door. Rocking back and forth from here and threatening sweeps to the front and back (and coming out to their back either way) seems to be the key to staying aggressive and not getting squashed here.