How can I modify this circuit to if once the battery turned off, never turn on again or turn only when min. ~0.5V increased?
It sounds like you need an under-voltage-lock-out (UVLO) circuit like this: -



You would select R5 to provide the right amount of positive feedback to ensure the switching points are at 3.0 volts and 3.5 volts. So, once the MOSFET is turned off, U1's output raises the target voltage at point B and this means the Li-ion cell has to "recover" a certain amount before the circuit reactivates the MOSFET and applies power to the load. Once power is re-applied, the op-amp output is low thus setting a lower target voltage at point B hence, the Li-ion has to drop to a lower threshold before the load is disconnected.

You might also be interested in the TLV3011/3012 - it has a voltage reference built in and can be "wired" to have hysteresis.