Signal deconvolution from exponential and linear tail
You probably can't. If I interpret your description correctly, this is the unit step response of your system. This looks very much like the unit step response of a high pass filter, that means the system is filtering out exactly the frequency that you want (which is DC or 0 Hz).
You may be better off creating a parametric model of your system response and try to do a weighted least square error fit of the data.