A very common movement impairment is using the lower back to compensate for a lack of overhead shoulder mobility.
Here is a person with their hands above their head:
BUT, the person is actually leaning back to get this overhead motion:
Their hands are over their head, but, relatively speaking, they are not over their torso. The torso is leaning back, but the arms are not:
(The other way to view it is neither is perpendicular to the floor.)
This can be a habit, but it's also often commonly from a thoracic (upper) spine that isn't extending effectively. If the upper back doesn't have the motion, the body tries to get it another way => extend the lower back.
Image source and more detail.
If you take these people and have them perform an overhead motion with their entire spine against a wall, they will often have a rude awakening for how little overhead mobility they actually have. Again, not always. Some people it's simply a habit they've gotten into. However, when pain or stiffness is involved, usually that thoracic spine needs some work.
Example of overhead motion with back on wall: https://www.youtube.com/watch?v=qgc-QxIStyc
Lastly, I'm a big fan of using the wall / support because the primary purpose of overhead pressing is to work the shoulders, not the lower back. Furthermore, by leaning back you start turning the motion into more of an incline press. Plus, if we can work on extending the upper back -which many need due to hunching over a computer all day- we get some more bang for our buck.