Use transistors to amplify a very small voltage
Yes, you can do this with just NPN transistors. 60mV is small, but probably enough.
Resistor values will depend on your voltage source, but try this:



simulate this circuit &ndash; Schematic created using CircuitLab
In normal conditions, R1 and R2 set Base voltage about 0.57V - not quite enough to conduct. Adding 30mV gets you 0.6V. With no Emitter resistor, you should get a decent current through R3.
You can also look at &quot;super alpha&quot; designs for very high gain. One example is here
EDIT:  there should be another resistor (RD) in series with D1. Just as R1/R2 bias the transistor in cutoff, RD would limit the voltage across, and current through, D1.
The last time I built this circuit, many years ago, I used a Light Dependenet Resistor (LDR) instead of a diode. LDR's have a resistance of Mega ohms in dark, dropping to about 10K in bright light, so voltage across them is not as big an issue. The same can't be said for diodes.