Circuit for detecting signal inversion
The slope being too shallow is basically solved by "amplification" but there are techniques you can do that will significantly improve your detection. Firstly, don't run the two outer loops as shorted turns, wire them antiphase so that the net voltage signal produced is zero when placed dead centrally straddling the centre wire.

This is called an inductive balance and now you are dealing with voltages and not trying to measure currents. This technique is used in professional metal detector equipment.

Next, parallel tune the output to give you: -


A decent voltage amplification
A decent out-of-band noise rejection


Then amplify to give you a decent signal level. Next, use a synchronous rectifier to convert the signal to a DC level that is zero in balance, positive when pushed in one direction and negative when pushed in the other direction.

If this sounds at all familiar it's because this is how linear variable differential transformers work: -



I would also tune the current carrying conductor to maximize current flow for the given DC power available. Again, something that is done in metal detection equipment to improve sensitivity.