Does the Corsair HS65 Surround need an amp? The spec sheet says 32k Ohms
Yes. The KA13 drives the HS65 Surround to about 115.3 dB, 5.3 dB above a 110 dB peak. A separate headphone amp is not what this pair is missing.
The FiiO KA13 has enough voltage for the Corsair HS65 Surround. Turn it up for the recording you are playing, not to make up a shortfall that is not there.
The HS65 Surround is 32 Ω. Reaching a 110 dB peak — a loud moment in film or classical, not a casual average — takes about 0.63 V from the source. The KA13 gets to 115.3 dB on this load.
If the tone sounds wrong, look at output impedance before you look at another amplifier. An amp upgrade here buys headroom you will not use and, on a sensitive headphone, sometimes a higher noise floor. Leave the volume where the music is comfortable. The peaks in a normal file are already inside what this output can swing.