There seems to be one very important aspect of the amplifier that hasn't been mentioned much here....
There are three reasons why using a "speaker amplifier" may sound better with particular headphones than using a "headphone amplifier":
1) If you have something like HiFiMan 6's, then their efficiency is very low and you simply need the power to drive them properly. If that's the case, then you probably don't need to especially worry about noise floor, and you'll have plenty of range on the volume control, so you won't be wanting any sort of attenuators.
2) Maybe it's simply a matter of the specific amplifier just plain sounding better....
3) Maybe it sounds better because your headphones really prefer having a low source impedance. Most (but not all) headphone amps have a somewhat high output impedance, which tends to result in a not especially flat frequency response, and one that varies depending on what headphones you use. Like speakers, most headphones present a rather complex load impedance to the amplifier - this complex impedance reacts with the output impedance of the amplifier to produce what can be very wide variations in frequency response. Some headphones are also sensitive to the amount of damping they get from the amp. (Just like speakers, damping allows the amp to "control" the movement of the diaphragm, and so possibly reduce distortion and other nonliniearities like ringing.) The result is that a given pair of headphones will sound quite similar on any amplifier with a low output impedance, but may sound very different on different amplifiers if their output impedances are high enough to interact with the impedance of the cans.
If you want to avoid confusion and poor results, it's important that you figure out which of these applies to you....
Here's why:
Most "normal speaker amplifiers", including the Mini-X, basically have a fixed-gain stage that is fed by the volume control - which is a variable attenuator. Since speaker amplifiers are designed to be used with speakers, they tend to have a much higher noise floor than, say, a headphone amp designed to work with sensitive IEMs.
If 1) applies to you, then your headphones pretty much act like speakers; they need the power, and aren't so sensitive that the noise floor is an issue, so its all good
If 2) applies to you, then you may or may not have issues because of the noise floor of the amp and because, with normal sensitivity phones, it plays so loudly that you barely get to move the Volume knob off the end stop.
If 2) or 3) applies to you, then you may be hearing the noise floor, and you may have trouble because the volume control doesn't "go down far enough"..... and, if so, then here's "the deal"...
If you put attenuators on the INPUT of the amp (the Mini-X), then you will be attenuating the input signal level. This will reduce the output level for a given input level, and so let you run the amp with the volume control turned up a bit more - which will make it easier to control. It will also attenuate any noise that happens to be coming from your source. And it
WON'T RAISE THE OUTPUT IMPEDANCE OF THE AMP AT ALL. However, it also won't do anything about the noise floor at the output.
If, instead, you put an attenuator on the OUTPUT of the amp, then you will be attenuating the output level; however, you will also be RAISING the output impedance. The result is that you'll be able to use the volume control set further up, and you WILL reduce the output noise floor, but you will be sacrificing the super-low output impedance.
I'll toss you a few representative numbers here....
The output impedance of a "typical headphone amplifier" can range from about 0.1 ohms or less to as much as 100 ohms or so.
(The output impedance of the headphone outputs on our DC-1 DAC is well below 1 ohm; many small headphone amps are around 10 ohms; most receiver outputs tend towards that high number).
The output impedance of a Mini-X is on the order of 0.02 ohms or so (the actual number depends on how you reference it - but it's
REALLY low).
The output impedance of a Mini-X with an attenuator on its
OUTPUT, as seen by the headphones, will depend entirely on the values you choose for the attenuator.
Specifically, if you use a "basic two resistor voltage divider" - with one resistor between the output and the load, and a second resistor to ground, then the output impedance will be the parallel combination of the two resistor values you use - in series with the output value of the amp itself. Either way, by doing so you will be raising the source impedance, and so reducing the damping and increasing the interactions between the amp and the headphones.
For example , if you use a divider composed of a 100 ohm resistor in series, and a 10 ohm resistor to ground, you will raise the output impedance of the amp (as "seen" by your cans) from about 0.02 ohms to about 9 ohms. This is a significant difference, and quite possibly a high enough output impedance to result in audible interaction with your headphones - both in terms of non-flat frequency response and reduced damping. If, instead, you use a 22 ohm resistor in series, and a 2 ohm resistor to ground, then your overall output impedance will be about 1.8 ohms.
You'll get the same 10:1 voltage reduction with either one, but that 22 ohm resistor will have to handle significantly more wattage. However, if your headphones are medium impedance (say 32 ohms or 50 ohms), the lower values will give you a lower source impedance, which may well give you an audibly better frequency response due to reduced interactions with the phones, and may also give you cleaner sound due to the better damping. (Of course, if noise isn't a problem, it's best of all to connect the phones directly to the amp's outputs.)