As mentioned, the current at the instant of opening the circuit is the same current that exists just after that.
Then the current decays, as energy is used up in the resistance of the coil, and the forward voltage of the diode. The diode holds the voltage to the supply voltage plus a diode drop (whatever it is at that current). This is because "X" amount of "volt seconds" of energy is stored in the inductance (1/2 x L x I^2 )
The diode needs to easily handle the current for the time it takes to decay. They are cheap enough that in most cases it costs more to spend time optimizing to find the smallest diode that can be used. Decay time is important for the relay release time, which may be an important issue, however.
To improve decay time, a small resistor can be added.... size that so that (R x I) plus the diode drop does not exceed your safe voltage. The resistance will speed the opening of the relay etc, by removing energy faster. (it is volt-seconds again.....The volts are made higher during the dissipation time, so seconds are smaller, the energy is removed faster)
The capacitor idea is actually good for ignition, but bad for relays. First, it increases the turn-on surge of the switching transistor, which has to charge the cap and supply coil current. Also, it stores even more energy, and that needs to be dissipated before the relay opens.
It can be done by combining the capacitor with a resistor, forming a standard arc damper. But BOTH are needed, and you need to consider the "Q" of the resonance, to make sure the energy is removed quickly. The resistor does the dissipation, adjusting the "Q" (sharpness of resonance and time to dissipate energy) to allow a quick relay release. Diodes are usually less trouble and often cheaper, both in cost and in PC board space. Years ago, capacitors were cheaper than the rectifiers, and were commonly used. (Still are where diodes are not appropriate, as in AC circuite). MOVs are good if you can tolerate the voltage, because they increase the volts, and so decrease the seconds, the time to release the relay.
In an ignition, you want to KEEP the energy IN the coil core, and redirect it to the secondary to make a spark. The capacitor does that, and allows a re-strike spark, since there is a resonance, so there is a chance to have multiple sparks. In the process, the voltage at the points can reach several hundred volts in a 12V system, repeatedly. This is NOT what you want in a relay coil, where you want to quickly dissipate the energy (volt seconds) that exists in the relay coil, and limit the voltage.