How can I tell if I need a better clock for my DAC?

A clock is such a small part of the big picture . I have found 
if you have a external power supply this is vastly more profound an upgrade. The  Mojo  Illuminati -2 external power supplies are miles ahead of what are stuffed in the stock boxes.
i have a Lumin dac- D-1 Player  that comes with a crappy smps ,
evrn a Sbooster Linear supply is much better. The Mojo is on par 
even the Lumin flagships S-1  Power supply  at 3x the moneys .
my point is if you can improve your power supply well worth the efforts.

This Stereophile AES J-test includes a very high level signal mixed with a LSB (smallest signal) and is a great test for interface jitter and any modulation distortion.

Ex 1. Benchmark DAC3 HGC, high-resolution jitter spectrum of analog output signal, 11.025kHz at –6dBFS, sampled at 44.1kHz with LSB toggled at 229Hz: 24-bit TosLink data (left channel blue, right red). Center frequency of trace, 11.025kHz; frequency range, ±3.5kHz.

https://www.stereophile.com/images/1117BDAC3fig11.jpg

There is no jitter (spurious signal) visible above -150 dbfs noise floor on the analog output. This means there is excellent interface jitter rejection.

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Ex 2. Schiit Yggdrasil, high-resolution jitter spectrum of analog output signal, 11.025kHz at –6dBFS, sampled at 44.1kHz with LSB toggled at 229Hz: 24-bit USB data (left channel blue, right red). Center frequency of trace, 11.025kHz; frequency range, ±3.5kHz.

https://www.stereophile.com/images/217Schiitfig12.jpg

There is jitter (lots of low level spurious signal at very specific tones).... probably inaudible but it is there.

Get an asynchronous DAC! See kijanki's post from 2-18-2018! Problem solved!