You can only have a single time constant with one capacitor, as Tc is fixed due to using a fixed capacitor and fixed (very low due to wiring) resistor.
Could you draw a quick schematic of how to achieve what your saying with a single capacitor. Thanks
All conventional electronic knowledge points to T = RC. T, the time constant, is the time taken to charge the cap to 63.2%., and it takes roughly 5T periods to consider it almost fully charged
With a single capacitor , in a fixed circuit, the time constant cannot be changed, and the time constant remains the same for both rising (charging) and falling(discharging) voltages
It might be that charging and discharging resistance is different, like e.g. smoothing caps in a power supply. They are charged to voltage in a few msecs, but may hold quite a high voltage for minutes...
Ths may actually mean that you have different time constants.