What load are you driving?
How much current do it need to produce?
Does it need to sink or source current?
What accuracy do you need on the voltage levels?
If you can try several 555 timers and select one with the lowest output voltage ..
According to the 555's data sheet(s) the output "L" voltage should be below 0.1 (mav 0.25V) for 10mA load but can rise to 2V (max 2.5V) for >100mA load ..
I thought you were concerned that the 555's output didn't go as low as 0V ..
The "subject" of you post didn't reflect clearly what you had been trying to achieve ..
If you would like to "shift" the "L" level to 0.3V you may consider connecting 555's GROUND pin to 0V through a Schottky diode ..
For example, the 1N5821 Vf voltage is somewhere around 0.3V ..
This should work. Be aware that positive spikes on the supply (which are common in automotive systems) will get rectified by D1 and C2, causing your output voltage to "jump" when one occurs. This may not be a problem.
Also - the LED that was removed (D2) would have been so dim as to be useless. You can add a separate resistor and series LED from pin 3 to GND if you want.