So to improve overhead strength, I like to work on several different aspects. Most of my lifters either lack absolute strength overhead, reactive strength,starting strength or have mobility/ flexibility issues (Note: I like to categorize strength based on Verkoshansky's seven types of strength [click here]: Types of Strength)
How do I evaluate where the lifter is or what they lack?
Mobility- I use many different tools, but I basically look for the ability to have more or less neutral shoulder position, scapulae pulled down then together, and palms flat overhead in a snatch grip.
Exercises to improve mobility: There are many, too many to mention here. I do suggest working with experts on this one, including Mobility Doc and Nelson Chiropractic and Pilates
Absolute Strength: People are weak. I have some strength standards I like to use: If someone can push press 80% of their jerk, they are strong (total arbitrary judgement based only on my experiences). If they can't, we can make them stronger and get good gains in their overhead strength. jerk/ snatch. I assume they can jerk what they can clean, and I also assume they can jerk with decent form a decent amount of weight (at least bodyweight or more). A second assessment I like to use if if people can do a jerk recovery with 150% of their jerk. Again, it's an arbitrary assessment; I don't know if this number is the right one to use or not, but I see people who don't have problems overhead and they can do this, and I see people with problems overhead who can barely do a jerk recovery with 90-110% of their jerk.
Exercises to improve it:
Incline Bench Press is the single greatest, and most often overlooked, strength developer in the cannon. People just don't do this enough anymore. People have returned to benching, but not inclining. Barbells, thick bars, dumbbells, alternate dumbbell, whatever. I like a higher incline, 60 degrees, but 45 is fine. 30 or so reps, more if the athlete needs shoulder and chest mass. Sets?: 5x5, 6x6, 4x6, 5-5-5-3-3-3, etc. I am a big fan of 3x3 at a heavy set for these for myself with a couple warm up sets at 5-8 reps.
Push Press+ Jerk (split or power) teaches leg drive and coordination. Snatch or clean grip, it's helped one of my lifters a ton and several of my others a lot.
Jerk Recoveries: holy crap these are awesome. 1x a week for 6 weeks or so work up to heavy doubles EMOM. Finish the last week, or week 7, with a max test. These can be done as an extra workout and are fairly easy to recover from. Repeat every 12-24 weeks.
Bench Press: I like using a thick bar or axle these days to protect people's shoulders, but benching works. Feet flat, back arch (not excessive, but you should be able to make a fist under the athlete's back), no butt movement, focus on pulling the bar to the chest and exploding off. Also dumbbells, alternate dumbbells, etc
People lack the ability to move weight from a given position without using the stretch-reflex; sometimes this is also a part of their reactive strength failures
Sots Presses and Snatch presses from the Squat- I never use actual Sots presses in training people, but I do use Kettlebell Sots Presses and Snatch presses from the Squat; they really teach engagement and force muscles to fire in the proper order. Completely humbling exercises; same rep range as the incline bench
Lock Outs: Set up pins or safeties in a rack at eye level, press from there; same rep scheme as the jerk recoveries
Floor Presses: Lay on the floor and bench off pins in the rack or safeties; your elbows should be 90 degrees; same rep scheme as jerk recoveries
Dumbbell Bench with a light band around your back: Grab the dumbbells and string a light band around your back, so that when you press you stretch the band out. Sets of 3-5, 5-10 sets.
You can tell if people have bad reactive strength overhead if their snatch or jerk gets into position for a second and they drop it by losing tension they almost had.
I like training overhead reactive strength through a series of movement types, not necessarily specific movements. My favorite is power snatching or push jerking with a pause at the catch. Also, I like slowly lowering under weight. We start with those two types and move toward the third type.
Similar to the first two exercises is the drop snatch, or snatch balance with no counter movement: start in pulling foot position and just drop down by lifting your knees. Catch the bar in a deep overhead squat with no wiggling or moving around. You can also do these into a split jerk or push jerk
Pressing snatch balance: start in your receiving foot position, bar on your back, snatch grip. Push yourself down into an overhead squat without letting the bar rise above the starting point. Just push under. Slowly do this to a five or six count (four if you're short).
Do all of the above (reactive strength) stuff with kettlebells for mobility and flexibility and unilateral work as needed.
Counter-Movement Movements for overhead reactive strength:
Snatch Balance, or heaving Snatch Balance as I was taught to call it in 1993. Also, the split jerk or push jerk.
Flexibility I define, for my purposes, flexibility as the ability of the ligaments and tendons and muscles to support a joint through a desired range of motion. So, for much of my purposes, flexibility and strength are nearly the same thing. To train flexibility strength for overhead I like the following:
Landmine presses from the knee- teaches people to engage the hip drive and put shit overhead.
Sots presses w/ KB and SN presses from squat
Dumbbell Pull Overs (mainly for adolescents, sometimes for older people)
Handstand walks and free standing handstand push ups- work toward these as a skill
Train forearm and tricep strength, and biceps to hold the stuff overhead. 2 exercises 1-2 x per week for reps is all you really need; train triceps, forearms, and biceps separately.
The back is also an integral part of overhead training, but back strength is better addressed as a part of the discussions on why people can't pull properly.