Resetting 555 via Transistor - RESET PIN LOW TOLERANCE
The very simple answer here is that you should be using an NPN transistor to pull the RESET pin to GND. With the circuit that you have with the PNP the emitter will always be ~0.65V higher than the base. Even with the base all the way to GND the emitter would still be about that 0.63V that you are seeing.

Using a proper circuit with an NPN if you want to keep the RESET asserted low you just apply current to its base through a resistor from a positive voltage level greater than about 1V. To allow the RESET to go high you would then bring that base resistor input to the NPN transistor to near ground.

NPN transistors that would work for you here could be a 2N3904 or a 2N2222.