It is not necessarily a ground loop.
It could be stray fields picked up by the coil. Think switch mode power supplies. Why no 60Hz? Because there are no 60Hz magnetic fields around nearby? Alternatively it is there, but from two sources (transformers) that cancel out. However moving the coil would show that up. It may be getting filtered out by a sampling artifact (notch or comb filter). This is possible if you haven't addressed anti-aliasing filter before the ADC.
I expect you are using a fourier transform to analyse the signal, so the noise could be due to sampling, aliasing etc. The sample rate should be something like 1 kHz with a practical order anti-aliasing filter. The ADC used to sample the signal could be getting noise from a ground loop or even capacitive coupling with digital parts of the system that are also running a program that goes at a rate related to the ADC sample rate).
Eliminate a ground loop from the signal driving the coil by using two wires attached to the driving circuit only, so there are no other shared signals in PCB tracks. One way is to use a transformer. Or just improve the wiring layout by using star grounds. A ground loop occurs when two different signals use the same wire (usually a common ground area on a circuit board). There is voltage drop (proportional to the currents) and this is in series with both signals. By keeping the signals confined to separate circuits, even if they are commoned at a single point, there is no ground loop. That is the principle of star grounds, everything returns to the same point, so there is no shared current path.
The same ideas apply to ground loops at the receive direction from the coil. You can use a transformer or differential amplifier (instrumentation amplifier) or an isolation amplifier to break the ground points.
Resolution...
It may be that you are just using too much resolution, so that noise becomes visible. It is unlikely that noise will be much less than 0.01% of the full scale where there is noisy digital circuitry nearby. That means about 12 bits. The random noise from ordinary op-amps etc is not up to 16 bit performance. Make sure the signal to noise is good at the ADC input. Your signal peaks should be not less than half the ADC full scale. Check them out with an oscilloscope.
Sampling rate...
If you want to do DSP such as fourier analysis the signal needs to be precisely and regularly sampled within the resolution of the ADC. This cannot be left up to software loops unless you are very careful. More likely needs clock triggered ADC with interrupts. While we all learned sampling needs only be twice the maximum frequency there are some serious provisos. There is no such thing as an anti-aliasing filter to do that except at very low resolution. For example a 12 bit ADC would need a filter with slope of 72dB per octave to eliminate all possible aliasing. It makes sense (and in this case should be easy) to go 10 times or more faster than the highest frequency of interest. The software can average this if you want less samples.
Aliasing...
The signal needs to be filtered before being sampled. The best filter is a Butterworth low pass filter for flatness and minimum phase annoyance. The cutoff is just above the highest frequency of interest. A fourth order filter is about the highest that can be easily achieved (using active filters). The slope is 24dB per octave. A 12 bit ADC has a dynamic range of 72dB. The filter needs to be 72dB down by the sampling rate. This takes 3 octaves, so if 100Hz is the cutoff, 800 Hz to 1KHz becomes the sample rate. Use good capacitors if you build one, and check its performance.
Hope this gives you some ideas, as I can only guess what you are doing.