Did you try messing with the switches on the top or bottom of the gamepad? One of them turns the extra buttons into turbo versions of button 1 and 2, which ends up disabling 3 and 4. I have used all 4 buttons with success in Wolf3D.

The answer is no, and I will try it when I have the chance. But, on my manual, it stated both switches were for switching between left/right handed modes.

The pad was purchased used, and the previous owner must have set the switches out of position. By tinkering with different switch settings, I got it working on 4 buttons. Thank you.

Maybe I can provide further helping by quoting the paragraph from the wiki :

Both at the top and bottom of the gamepad are switches. One of them removes the normal functionality from 2 of the buttons, and turns them into autofire variants of the first 2. This gave all four buttons functionality even in PC games that only supported two buttons on joysticks or for scenarios when two gamepads are connected with a Y-splitter. The other allows for left-handed operation by turning the workings of the D-pad and buttons upside down. Both switches can be used at the same time.

That explains it. I should've checked Wikipedia instead of the little insert that came with my controller. Thanks.