The numbers I am looking for seem to be exclusive to "Unadilla" class gunboats. In some cases, they are painted on the stacks of the boats, but either that is not the case for every boat, or else the numbers just don't show on the available pictures of some boats.
They might have been used throughout the war, or maybe not. Once the tinclads arrived in the lower Mississippi, the Unadilla class boats seem to have left (correct me if I'm wrong on this). But with the tinclads all having designated numbers, it seems it would have been very confusing for the Navy to have had more than one "Gunboat #7" and "Gunboat #8", etc., running around, especially on the same stretch of river.