How to measure 1/f noise close to DC
You say apply DC current to "a material" and this suggests to me the possibility of using more than one independent block of material...

The essence of your problem seems to be how to apply a high DC gain amplifier without the (naturally, DC) bias overloading the amplifier - and one approach suggested in comments is to apply a DC bias : the objection to that is that the DC bias source is also a 1/f noise source, and so provisions must be made to compensate for that somehow.

Another classic approach is a "bridge" configuration in which you acknowledge the DC bias is another noise source - but make it of the same material with an identical current source (or a current mirror) - and amplify the difference between the two DC voltages. Then compensation for two (presumably) independent noise sources is purely mathematical : adding 2 uncorrelated random (both assumptions!) noise sources adds 3dB to the noise level, i.e. doubling the noise power, or raising the noise voltage by sqrt(2).

There is a variety of amplifier called a "chopper amplifier" which converts these two DC inputs to AC and amplifies the resulting AC signal, thereby eliminating any 1/F noise contribution of its own. Probably useful here...