If you want your verb form to convey, explicitly and unambiguously, the idea that you had the realization immediately after the car had (fully) started, and not rely solely on adverbial phrases to convey this idea, then you would use the past perfect:
As soon as I'd started the car, I realized that...
No sooner had we spread the picnic blanket out on the grass than it began to rain.
As soon as we had spread the tarp on the field, the rain stopped.
The spreading of the blanket on the grass was completed and then it began to rain.
The simple past, while grammatical, does not explicitly convey the idea that the action had completed. Rather that meaning is implicit in the statement in its entirety, and the meaning is not always unambiguous.