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For noise work we had multiple Faraday-cage "boxes" around probe chuck or socket, and these had high-zoot LNAs inside the cage before any instruments. A well designed gain block would help your DMM "effective range" though fractional resolution "is what it is". Then a lot of oversampling to drive down ambient and instrument internal noise floor. Triax cables are the done thing for DC leakage and could "double shield" AC incoming EMI. Line sync can help consistency though not accuracy@sync-point necessarily (crossing vs crest). If not synced then be sure to accumulate across many line cycles at many points per, for "unbiased" averaging.