Adjustable 555 timer off duty delay
If I'm reading this question  correctly you need two monostable circuits. 



The circuits are both negative edge triggered monostables. 

The first negative edge comes from the switch, S1, (this could be replaced by an NPN transistor if you want it to be triggered by another device). C1 and R2 produce a short spike from +V to 0V on the negative going edge of the input signal. This starts monostable 1 with R3 and C2 controlling the length of the pulse (T1) When the switch is opened diode D1 limits the positive going edge spike amplitude to 0.6V above +V, but this will not retrigger the monostable. 

When the first monostable goes high R4 and C2 only allow the positive edge of this pulse through. D2 limits this to 0.6V above +V. This positive edge will not trigger the second monostable.

When the first monostable output goes LOW the negative edge of the pulse triggers the second monostable and its output goes HIGH for a period controlled by R5 and C4.

By having two separate monostable periods you can independently adjust the initial delay (T1) and the operating pulse time (T2)