I tried a version of this without any improvement. Maybe I got it wrong. I will repeat the experiment.Just go with increasing the noise gain as OBW5409 suggested in post #14.
It is nearly fool-proof and has a very high success rate.
If you find that the lower open loop gain is affecting DC accuracy you can install a capacitor in series with the shunting resistor (500 ohms in the given example). Use C = 1/(2 Pi R F) to find the capacitor necessary to have the gain reduction kick in at your desired frequency.