In this paper, we pursue the study of the problem of controlling the maximal singular integral T * f by the singular integral Tf. Here T is a smooth homogeneous Calderon-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the norm and via pointwise estimates of T * f by M(Tf) or M 2 (Tf), where M is the Hardy-Littlewood maximal operator and its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article, we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak (1,1) estimates, which are no longer true for odd kernels. For odd kernels, we obtain sharp weaker inequalities involving a weak L 1 estimate for functions in L log L.