Purpose of assistance exercises for main lifts is to work on the particular range of motion which is potentially your weakest link. This way you will solve the problem, and advance in your main lift. At the beginning of strength training your weakest link is everything, so you do only main exercises, until your weakest link show itself. For example, if your weakest point in the Conventional deadlift is your grip then you will do some extra grip work, like Timed grip and Finger hold. If it is lower back that is lacking, then the right choice is to do some specific lower back work, like Back extensions or Good mornings. Maybe it will be your hip drive at the beginning of the movement. Then the Box squat with Conventional deadlift stance is the lift that you will need to specialize. If you prefer Sumo deadlift, then it is Box squat with sumo deadlift stance, and so on.
Front squat, Paused squat, Box squat with squat stance, Bottom position rack squat, Belt squat, One leg squat (another leg behind, another leg to the side, or another leg in front).
Box squat with deadlift stance, Upper deadlift (Rack or Block pull), Vertical barbell row, Front barbell shrug, One leg deadlift, Back extension, Reverse back extension, Good morning, Grip hold, Finger hold, Glute-hamstring raise, Lunge.
Bottom position rack bench press, Bench press with paused reps at the bottom (3-5 sec.), Dumbbell bench press, Board press, Band bench press, Floor press, One hand floor press, Close grip bench press, Dips, Bench press with legs on the bench.
One arm dumbbell row, Sitting bent-over row, Bench row, Vertical barbell row, Machine row, Lat-machine pullover.
Pin press (from eye level to a lockout), Bottom press (from down to eye level), Upright row, One hand standing press, Sitting dumbbell press, Behind the head triceps press.
Partial pull up (from bottom to eye level), Bottom pull-down (from eye level to chest), Chin up, Hammer grip pull up, Pulldown, Lying pullover.
Great tool for additional strength gains for intermediate lifter could also be “small” exercises. For example, Calf raises in all variations are very important for strong and stable stance in Squat and Deadlift. We could also easily neglect arms in strength training programs, and I strongly advise you not to do that. Although doing Chin ups, Pull ups, Hammer pull-ups, Dips, Close grip bench presses, Overhead EZ or Dumbbell presses for upper arms, and Grip and Finger hold for your forearms will give you strength and probably big arms, you may benefit from some curls, triceps extensions, and wrist curls.
There are many small exercises that you may explore in your intermediate stages that could put some weight on your main lifts. You could add them when you still feel energetic at the end of your training. Lying dumbbell triceps extensions could be very beneficial exercises for bench press lockout. Also, reverse barbell curls and wrist curls in all four directions can play a role in additional strength gains. You will benefit from all exercises that mimic a part of the main strength movement. So, don’t be afraid of doing Lateral raises, L-raises, and Hip fly’s, here and there. Those movements will not only help to build more strength for your main lifts but will also help to keep your shoulders and hips injury free.