How to Use Shoulder Pressure
One thing changes the side control game perhaps more than any other: shoulder pressure. Imagine that you're on your back, controlled by your partner, but you're working hard on making frames so you can work on a textbook or unorthodox side control escape. Finally, after a few minutes of struggle, you get inside arm control, only to have it immediately taken away by the most crushing, brutal pressure you've ever felt (okay, maybe there is a tiny bit of hyperbole there, but maybe not). Shoulder pressure is, in short, a complete game changer for the top person working to advance their position.
Getting the Grip
One very useful strategy in overall control from the side control position is to first seek to immobilize the hips, then to move up to the shoulders. This is especially viable when you first pass the guard, and your partner may have some pretty solid defensive frames. Assuming the hips are under control, start by establishing a nice "head hug" position. If your partner is simply blocking your hips, you can just go for the head right away, but if they're trying to frame against your arm, you will need to do a small "swim" to get there. Once you are flush with your partner's head and neck, it's important to recognize that your partner's frame that is now on your hip is your main concern with regard to both nailing down your own submission, and also in regard to your partner's main escape route. Reach out at a 45 degree angle, give or take, and extend your arm so that it can reach much further across your partner's back. Next, use your right hand (assuming your left arm is under their head) to lift your partner's shoulder off the ground (slightly; this should be pretty easy, as they want to turn back in toward you in general). This will allow you to grip their far shoulder, right as the rear deltoid muscle, your target group, will be exposed.
The Head Turn
Note that, prior to getting the grip, your base is exposed if you select to switch your hips to deal with double defensive frames. That's why the shoulder pressure approach here is much better to use. Once you have established your grip on your partner's rear deltoid, it's time to turn their head away from you. If they can't look at you, they certainly can't turn in to escape with their hips, so you are eliminating the majority of your partner's escape routes by simply turning their head away. Next, use the "head turn" to slide your left knee under your partner's right elbow, forcing their arms into a terrible defensive posture (allowing you access to goodies like the triangle from side control). Note that you can either opt to choke your partner (if your arm is pinching across their neck), or just turn their head (very effective if control is the only objective).
Inside Arm Domination
It is important to remember that the main goal of attaining shoulder pressure is not merely to torture your training partner (that is just a bonus), but rather to use the opportunity of discomfort to slide your near-side knee in to gain control of the inside arm. Preventing any meaningful attempts at escapes means you can now focus on the attack, and, consequently, your partner will necessarily be on the defensive, unable to focus on anything other than survival. Here's a good look at the "windshield wiper" move to clear that arm.
Head turn or neck smash?
While there isn't a right or wrong answer to the eternal question: "How should I use my side control shoulder pressure - to choke, or just to turn the head?", there are definitely things to consider with each option. First, if you want to choke your partner with one arm, you most definitely can. If one carotid artery is cut off 100%, your partner will eventually be forced to tap or go to sleep. This is a huge benefit in and of itself for obvious reasons, as you can then set up other clean attacks like the arm triangle. On the other hand, turning the head is most effective and efficient when you focus not on their neck, but instead on their cheek and jawline. If you can turn your partner's skull, they literally will be unable to turn toward you. I'll leave the decision to you!