We present a detailed study of s-process nucleosynthesis in massive stars of solar-like initial composition and masses 15, 20, 25, and 30 M. We update our previous results of s-process nucleosynthesis during the core He burning of these stars and then focus on an analysis of the s-process under the physical conditions encountered during the shell carbon burning. We show that the recent compilation of the ²²Ne(α,n)²⁵Mg rate leads to a remarkable reduction of the efficiency of the s-process during core He burning. In particular, this rate leads to the lowest overproduction factor of ⁸⁰Kr found to date during core He burning in massive stars. The s-process yields resulting from shell carbon burning turn out to be very sensitive to the structural evolution of the carbon shell. This structure is influenced by the mass fraction of ¹²C attained at the end of core helium burning, which in turn is mainly determined by the ¹²C(α,γ)¹⁶O reaction. The still-present uncertainty in the rate for this reaction implies that the s-process in massive stars is also subject to this uncertainty. We identify some isotopes like ⁷⁰Zn and ⁸⁷Rb as the signatures of the s-process during shell carbon burning in massive stars. In determining the relative contribution of our s-only stellar yields to the solar abundances, we find it is important to take into account the neutron exposure of shell carbon burning. When we analyze our yields with a Salpeter initial mass function, we find that massive stars contribute at least 40% to s-only nuclei with mass A ≤ 87. For s-only nuclei with mass A > 90, massive stars contribute on average ~7%, except for ¹⁵²Gd, ¹⁸⁷Os, and ¹⁹⁸Hg, which contribute ~14%, ~13%, and ~11%, respectively.