So here’s a way to tell - without destroying anything - if the single-ended cable you already have is 4-wire, i.e. well structured to minimize crosstalk, or has a Y-shaped GND with a significant undesirable portion of common-GND. Measure these 3 resistances:
If R3=R1+R2, congratulations: it’s a healthy baby 4-wire. If R3 is significantly less than R1+R2 and your headphones are low-impedance (say, 32-ohm or less), you could benefit from getting a better cable, even while keeping your SE amp. (I just measured the stock HE-400i cable and it’s a healthy 4-wire, with 0.4-ohm GND wires on each side and 0.8 ohms when measured together in series from left to right; shows HiFiMan knew what they were doing on the electrical side of their cable build.)
LE:
Yes, for the particular cable in the picture you can already tell it has good structure because the portion below the joining point still looks like 2 independent 2-wire cables going all the way to the amp-side plug. 
The above method is for when you can’t see this structure from the outside.