The upper limit and the first gap in the spectrum of genera of F-q(2)-maximal curves are known, see [34], [16], [35]. In this paper we determine the second gap. Both the first and second gaps are approximately constant times q(2), but this does not hold true for the third gap which is just 1 for q equivalent to 2 (mod 3), while (at most) constant times q for q equivalent to 0 (mod 3). This suggests that the problem of determining the third gap which is the object of current work on F-q(2)-maximal curves could be intricate. Here, we investigate a relevant related problem namely that of characterising those F-q(2)-maximal curves whose genus is equal to the third (or possible the forth) largest value in the spectrum. Our results also provide some new evidence on F-q(2)-maximal curves in connection with Castelnuovo's genus bound, Halphen's theorem, and extremal curves.3233589608