To measure current through a wire I used a 324 quad op amp because spec said its output goes down to 0V. Input up to tens of mV. Gain=200.
The op amp had an unwanted offset. Zero input did not produce zero output. To reduce the offset, I had to contrive a way to nudge the inverting output downward a little.
So I made a resistor divider from 3 low ohm resistors. I arranged it to apply a tiny voltage between 0 and supply+. It was a big headache to find the right combination. I suspect that it hindered accuracy and/or linearity. I'm not sure the 324 was the best choice for the task. And I wanted a single-ended supply.
Most likely a proper solution is to install a negative supply. Raise or lower its voltage in a manner that affects the output up or down. The voltage can be adjusted directly, or with resistors, or a diode/zener/led. Or even by putting a pot across a diode, to dial a fixed voltage drop. Etc.