Straight-bar overhead pressing is a great diagnostic tool. It'll tell you a lot about your spinal stability, scapular mobility, shoulder strength and function. But it's rarely a good fit for those struggling to achieve good positions or those who have bad posture.
First off, the symmetrical stance (feet parallel on the floor) has the potential to create instability and hyperextension at the lumbar-pelvic junction. It's this lack of trunk and pillar support that becomes the weakest link in the functional chain. It limits top-end loads and places unwanted stress on the spine.
Since the body is a functional unit, this hyperextension can become more notable when you lack the ability to achieve a neutral, or possibly even slightly extended, thoracic spine position. A rounded mid-back will be tough on the lower back and also on the shoulder, reducing space in the shoulder girdle for structures to properly function.
Between the lack of lumbar spine stability, thoracic spine mobility, and the ability to display functional range of motion and stability through the shoulder girdle, the standard overhead press can be risky.
Do This Instead: Dynamic Scrape-The-Rack Press
You might be familiar with this one. Coach Thibaudeau goes into detail with it in Tip: Do the Strip-the-Rack Press. It lessens all the problems that come with traditional overhead pressing, and it's become a staple pressing movement for my athletes.
Using a split stance will reduce the hyperextension of the lower spine and pelvis. Additionally, the friction between the bar and the rack increases the stability through the shoulders, and it increases the centration of the shoulder joint itself. This stimulates the intrinsic stabilizers of the gleno-humeral joint and scapula making everything work together better.
Use a False Grip
If you want to take this press to the next level, try using a false grip. This will reduce the carrying angle of the upper arm and position the shoulders in a more naturally centralized position.
Since the anterior deltoid is largely comprised of fast twitch muscle fibers, train this movement explosively for low to moderate rep ranges. This will reduce the need to overload the exercise.