It all depends on the type of noise you are experiencing. If it's about high-frequency noise being aliased by the distortion stage, try applying an analog low-pass filter on the output of BelaMini, that would reduce the high frequencies components that may be affecting your distortion pedals. I'd start with a passive RC low-pass filter:
belaoutput-----R1----+-----pedal input
|
C1
|
GND
start off with R1=2.2k ohm and C1=10nF. This should give you a cutoff of 7.2kHz. If you experience considerable improvement this way, you can tweak the values to adjust the cutoff, which is given as:
1/(2π * R1 * C1)
(make the resistor smaller to increase cutoff).
If it's more of a hum, then possibly using a LiPo battery could help. At the very least, it would help diagnose whether there's a significant improvement to be had by using it.