I don't think adding C3 really gives you the behavior you want, and it presents some risks. You have to remember that when Q1 closes, the inductor will want to continue discharging current, this is the point of the fly-back diode, to allow a path for this current to discharge. At this point, the current flowing through the inductor raises pin A1 to a higher potential than A2, hence a positive voltage across the diode and a discharge. The diode limits the voltage, otherwise, without the diode, the voltage can increase to dangerously high levels.
C3 might not help you keep the relay open, as it's going to be charged in the opposite direction. At the moment Q1 closes, C3 will hold a positive charge across its terminals, but the inductor will want to produce a negative voltage to discharge its current. It won't, therefore, be able to discharge through the inductor, and you've created an LC tank that will possibly resonate, or possibly destroy the capacitor. I ran a quick simulation to show what I mean, note the starting conditions, Vcap = 12V, Lcurrent = 0.1 uA. The values are arbitrary but they demonstrate the point. The red curve is the voltage across R1, which is standing in for the closed transistor, and the green is current through the inductor. R2 and R3 stand in for wire losses.
In this instance, I get obviously a silly value of current, but hopefully, it demonstrates the point. In reality, you might be able to tune and damp this to get a more desirable behavior, but I wouldn't recommend it. [edit] The various values here are just arbitrary, and I'm not proving it will oscillate, just showing it could. Equally, you can get very high voltages with this circuit, which comes back to the need for a fly-back diode [/edit]
As for your intent, to add some time delay to the control to compensate for short dead times, I'd recommend conditioning your signal. I don't know exactly what your control signal is, so I can't give specifics, but I'm going to assume you're able to buffer it to some kind of digital format. One you're input is more or less digital, you can then condition it.
In the above image, D1, C1, and R3 produce the trace in green. C1 and R3 form a RC filter, D1 stops the input from discharging the circuit and allows the slow discharge through the resistor. Pass this to a comparator, compare it to another value and you've got a way or removing a small dead time from your signal. R4 could be replaced with a MOSFET to drive the relay, or you could directly drive if your op amp is beefy enough.
It's just then a matter of crunching the numbers and getting the times you want. You could replace R1&2 with a variable resistor so you can tweak this in situ.