We describe the first s-process post-processing models for asymptotic giant branch (AGB) stars of masses 3, 4 and 5 M at solar metallicity (Z=0.018) computed using the input from the stellar evolutionary code aton. The models are computed with the new code snuppat(S-process NUcleosynthesis Post-Processing code for aton), including an advective scheme for the convective overshoot that leads to the formation of the main neutron source, 13C. Each model is post-processed with 3 different values of the free overshoot parameter. Included in the code snuppat is the novel Patankar-Euler-Deflhard explicit numerical solver, that we use to solve the nuclear network system of differential equations. The results are compared to those from other s-process nucleosynthesis codes (Monash,fruity, and NuGrid), as well as observations of s-process enhancement in AGB stars, planetary nebulae, and barium stars. This comparison shows that the relatively high abundance of12C in the He-rich intershell in aton results in as-process abundance pattern that favours the second over the first s-process peak for all the masses explored. Also, our choice of an advective as opposed to diffusive numerical scheme for the convective overshoot results in significants-process nucleosynthesis also for the 5 M models, which may be in contradiction with observations.