Problem with wide voltage range CC sinking
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. A quick test.

I suspect that your problem is that Q1's emitter is one diode-drop below ground - maybe at -0.6 or -0.7 V. Meanwhile the transistor is in saturation so the collector is probably 0.2 V higher than the emitter at -0.5 V.

A quick test would be to raise the base voltage by one diode drop - done here by R2 and Q2. This should hold the emitter closer to 0 V and may be adequate for your problem. I've chosen to use the base emitter junction as a diode so that the voltage drops will match.