There are two hidden armor parts that X can receive.

The first is hidden in Doppler Stage 1, which is only available if X has found all items and has NOT taken any of the enhancement chips. The capsule contains a Hyper Chip that grants him gold armor and the effects of all the other chips.

The second is in Doppler Stage 2. If the player has killed Vile before this stage and has allowed Zero to survive, using Zero through the stage makes him engage a miniboss. The fight ends with him getting damaged, and he gives X his Z Saber before retreating. Without arm enhancements it works basically like how Zero uses it in game, but with the arm parts the slash becomes a flying beam like Zero uses in Mega Man X2.

The reason the boss cannot be fought when Vile lives is that the second Doppler stage is left in derelict condition, all the windows blown out by explosions. Presumably, this is because Vile, in his insanity towards X, did a tremendous amount of damage to the building before his arrival, and the flying miniboss escaped out the open window.