An important invariant of a chemical reaction network is its maximum number of positive steady states. This number, however, is in general difficult to compute. Nonetheless, there is an upper bound on this number -- namely, a network's mixed volume -- that is easy to compute. Moreover, recent work has shown that, for certain biological signaling networks, the mixed volume does not greatly exceed the maximum number of positive steady states. Continuing this line of research, we further investigate this overcount and also compute the mixed volumes of small networks, those with only a few species or reactions.