*Input* offset means the op amp can see a difference of >6mV *between* it's *inputs*. The output can still be over 6mV. You need a rail to rail opamp which goes down to zero volts at its *output*. I don't know if there's anything cheap which can do that.

The usual way to get a zero output is to use a balanced dual supply. And, then some kind of protection circuit to make sure output is within uC input limits. This makes every single op amp ever made go down to zero volts.

If you can find an opamp that does have 0 V output, but has a high input offset, you can still use the LMV324. Set a gain on the LMV324 and feed it into the other opamp.