I agree about 5/3/1 as a very solid option for increasing weight.
But for the overhead press, I personally got stuck at 0.7 Bodyweight with it.
To get to a Bodyweight Overhead Press, I needed more volume than standard 5/3/1. Maybe if I had practiced it everyday and often on top of 5/3/1 I could have squeezed more out of it.
But instead I did Pavel’s Enter the Kettlebell Rite of Passage.
It consists of getting to a 1/2 Bodyweight One Arm Clean and Press.
The progression works for barbells and dumbbells too without an issue, since they are simply ladders.
Ladders are also explained here:
An interesting barbell alternative is this Bill Starr routine geared for the Overhead Press:
More details about these 2 programs can be found on the internet.
Hope that helps