The minimum interpass temperature is addressed by D1.1. It must be at least equal to the minimum preheat temperature.
The maximum interpass temperature is typically not an issue when welding carbon and high strength low alloy steel unless notch toughness is a concern.
Think about how typical carbon steel and high strength low alloy steels are made. They are rolled while they are hot, allowed to air cool with no further heat treatment. Essentially, they are normalized, i.e., the are austenized and air cooled to ambient temperature. In some cases (most of the time) the steel surface is flooded with cold water to minimize the thickness of the mill scale. While it isn't considered to be a "full quench" because it isn't cooled with water below the austenizing temperature range, it is still closer to normalizing than it is quenching.
In general, if acceptable mechanical properties are developed by allowing the steel to cool from austenizing temperatures to ambient temperature in air; interpass temperatures in the same temperature range are probably no worse.
D1.1 lists those (few) base metals where the maximum preheat and interpass temperature must be observed in the notes/footnotes of Table 3.2.
Best regards - Al